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+{"text":"\\section{Introduction}\nThe underlying graph of each digraph in this paper is assumed to be simple unless otherwise mentioned.\n\nGiven a digraph $D$, the {\\em competition graph} $C(D)$ of $D$\nhas the same vertex set as $D$ and has an edge between vertices $u$ and $v$\nif and only if there exists a common prey of $u$ and $v$ in $D$. If $(u,v)$  is an arc of a digraph $D$,\nthen we call $v$ a {\\em prey} of $u$ (in $D$) and call $u$ a {\\em predator} of $v$ (in $D$).\nThe notion of competition graph is due to Cohen~\\cite{cohen1968interval} and has arisen from ecology. Competition graphs also have applications in coding, radio transmission, and modeling of complex economic systems.  (See \\cite{raychaudhuri1985generalized} and \\cite{roberts1999competition} for a summary of these applications.)\nVarious variants of notion of competition graphs have been introduced and studied (see the survey articles by Kim~\\cite{kim1993competition} and Lundgren~\\cite{lundgren1989food} for the variations which have been defined and studied by many authors since Cohen introduced the notion of competition graph).\n\nThe notion of $m$-step competition graph is one of the important variants and is defined as follows.\nGiven a digraph $D$ and a positive integer $m$, a vertex $y$ is an {\\em $m$-step prey} of a vertex $x$ if and only if there exists a directed walk from $x$ to $y$ of length $m$. Given a digraph $D$ and a positive integer $m$, the digraph $D^m$ has the vertex set same as $D$ and has an arc $(u,v)$ if and only if $v$ is an $m$-step prey of $u$.\nGiven a positive integer $m$, the {\\em $m$-step competition graph} of a digraph $D$, denoted by $C^m(D)$, has the same vertex set as $D$ and has an edge between vertices $u$ and $v$ if and only if there exists an $m$-step common prey of $u$ and $v$. The notion of $m$-step competition graph is introduced by Cho~{\\em et al.}~\\cite{cho2000m} as a generalization of competition graph.\nBy definition, it is obvious that $C^1(D)$ for a digraph $D$ is the competition graph $C(D)$.\nSince its introduction, it has been extensively studied (see for example \\cite{belmont2011complete,cho2011competition,helleloid2005connected,ho2005m,kim2008competition,park2011m,zhao2009note}). Cho~{\\em et al.}~\\cite{cho2000m} showed that for any digraph $D$ and a positive integer $m$, $C^m(D)=C(D^m)$.\n\nFor the two-element Boolean algebra $\\mathcal{B}=\\{0,1\\}$, $\\mathcal{B}_n$ denotes the set of all $n \\times n$ (Boolean) matrices over $\\mathcal{B}$. Under the Boolean operations, we can define matrix\naddition and multiplication in $\\mathcal{B}_n$.\nA graph $G$ is called the {\\em row graph} of a matrix $A \\in \\mathcal{B}_n$ and denoted by $\\RRR(A)$ if the rows of $A$ are the vertices of $G$, and two vertices are adjacent in $G$ if and only if their corresponding rows have a nonzero entry in the same column of $A$.\nThis notion was studied by Greenberg~{\\em et al.}~\\cite{greenberg1984inverting}. As noted in \\cite{greenberg1984inverting}, the competition graph of a digraph $D$ is the row graph of its adjacency matrix.\n\nCho and Kim~\\cite{cho2004competition} introduced the notions of competition index and competition period of $D$ for a strongly connected digraph $D$,\nand Kim~\\cite{kim2008competition} extended these notions to a general digraph $D$.\nConsider the graph sequence $C^1(D)$, $C^2(D)$, $C^3(D), \\ldots, C^m(D)$, $\\ldots$ for a digraph $D$.\nNote that for a digraph $D$ and its adjacency matrix $A$, the graph sequence $C^1(D)$, $C^2(D), \\ldots, C^m(D)$, $\\ldots$ is equivalent to the row graph sequence $\\mathcal{R}(A)$, $\\mathcal{R}(A^2), \\ldots, \\mathcal{R}(A^m), \\ldots$.\nSince the cardinality of the Boolean matrix set $\\mathcal{B}_n$ is equal to a finite number $2^{n^2}$, there is a smallest positive integer $q$ such that $C^{q+i}(D)=C^{q+r+i}(D)$ equivalently $\\mathcal{R}(A^{q+i})=\\mathcal{R}(A^{q+r+i})$ for some positive integer $r$ and all nonnegative integers $i$.\nSuch an integer $q$ is called the \\emph{competition index} of $D$ and is denoted by cindex$(D)$.\nFor $q=$cindex$(D)$, there is also a smallest positive integer $p$ such that $C^{q}(D)=C^{q+p}(D)$ equivalently $\\mathcal{R}(A^{q})=\\mathcal{R}(A^{q+p})$.\nSuch an integer $p$ is called the \\emph{competition period} of $D$ and is denoted by cperiod$(D)$.\nRefer to \\cite{kim2010generalized, kim2015characterization, kim2012bound} for some results of competition indices and competition periods of digraphs.\n\nEoh {\\it et al.}~\\cite{eoh2020m} studied the $m$-step competition graphs of orientations of complete bipartite graphs for an integer $m \\ge 2$.\nThey introduced a notion of sink sequences of digraphs, which played a key role in the paper.\nGiven a digraph $D$, we call a vertex of outdegree zero a \\emph{sink} in $D$. We define a nonnegative integer $\\zeta(D)$ and sequences\n\\[(W_0, W_1, \\ldots, W_{\\zeta(D)}) \\quad \\mbox{and} \\quad (D_0, D_1, \\ldots, D_{\\zeta(D)}) \\]\n of subsets of $V(D)$ and subdigraphs of $D$, respectively, as follows.\nLet $D_0=D$ and $W_0$ be the set of sinks in $D$.\nIf $W_0 = V(D)$ or $W_0 = \\emptyset$, then let $\\zeta(D)=0$.\nOtherwise, let $D_1 = D_0-W_0$ and let $W_1$ be the set of sinks in $D_1$.\nIf $W_1 = V(D_1)$ or $W_1 = \\emptyset$, then let $\\zeta(D)=1$.\nOtherwise, let $D_2 = D_1-W_1$ and let $W_2$ be the set of sinks in $D_2$.\nIf $W_2 = V(D_2)$ or $W_2 = \\emptyset$, then let $\\zeta(D)=2$.\nWe continue in this way until we obtain $W_k=V(D_k)$ or $W_k =\\emptyset$ for some nonnegative integer $k$.\nThen we let $\\zeta(D)=k$.\nBy definition, $0 \\le \\zeta(D) \\le |V(D)|-1$.\nWe call $\\zeta(D)$ the \\emph{sink elimination index} of $D$, the sequence $(W_0, W_1, \\ldots, W_{\\zeta(D)})$ the \\emph{sink sequence} of $D$, and the sequence $(D_0, D_1, \\ldots, D_{\\zeta(D)})$ the \\emph{digraph sequence associated with the sink sequence} of $D$.\n\nIn this paper, we study competition indices and competition periods of multipartite tournaments in terms of sink sequences of digraphs.\nA \\emph{$k$-partite tournament} is an orientation of a complete $k$-partite graph for a positive integer $k$.\nIn particular, if $k \\ge 2$, then we call it a \\emph{multipartite tournament}.\n\nTwo vertices $u$ and $v$ in a digraph $D$ are\nsaid to be {\\em strongly connected} if there are directed walks from\n$u$ to $v$ and from $v$ to $u$. We say that a digraph $D$ is {\\em\nstrongly connected} if each pair in $V(D)$ is strongly connected. A\ndigraph $D$ is said to be {\\em primitive} if $D$ is strongly\nconnected and the greatest common divisor of lengths of its directed\ncycles is equal to $1$. It is known \\cite{brualdi1991combinatorial} that\nif a digraph $D$ is primitive, then there exists a positive integer\n$t$ such that there is a directed walk of length exactly $t$ from each vertex\n$u$ to each vertex $v$ (possibly $u=v$). The smallest integer $t$ is called the {\\it exponent of the\nprimitive digraph $D$} and it is denoted by $\\exp(D)$.\n\nIn Section~\\ref{sec:acyclic}, we deal with acyclic multipartite tournaments.\nWe show that the competition period of an acyclic digraph $D$ is one and $\\zeta(D) +1$ is a sharp upper bound of the competition index of $D$ (Theorem~\\ref{acyclic-digraph-properties}).\nEspecially, it turns out that the competition index of an acyclic $k$-partite tournament $D$ is $\\zeta(D)$ or $\\zeta(D) +1$ for an integer $k \\ge 3$ (Theorem~\\ref{Thm:acyclic k-partite tournament}).\nIn Section~\\ref{sec:directed cycle}, we handle multipartite tournaments with sinks and directed cycles.\nWe introduce types of a directed walk and types of a vertex in $\\bigcup_{i=0}^{\\zeta(D)-1}W_i$ where $\\left(W_0,\\ldots ,W_{\\zeta(D) }\\right)$ is the sink sequence of a multipartite tournament $D$ with sinks and directed cycles, and then show that each vertex in $\\bigcup_{i=0}^{\\zeta(D)-1}W_i$ is of Type~1 or Type~2 (Theorem~\\ref{thm:34cycle}).\nWe show that the existence of $(u,w)$-directed walk of Type~1 or Type~2 of a certain length for a vertex $u$ in $D_{\\zeta(D)}$ and $w \\in \\bigcup_{i=0}^{\\zeta(D)-1}W_i$ guarantees the existence of a $(u,w)$-directed walk of length $m$ in $D$ for infinitely many integers $m$ in a specific form (Lemmas~\\ref{lem:3cycle}-\\ref{lem:both34}).\nBy integrating these results, we show that the competition period of a multipartite tournament with sinks and directed cycles is at most three (Theorem~\\ref{thm:at most 3}).\nIn Section~\\ref{sec:primitive}, we show that the competition index of a primitive digraph is at most its exponent (Theorem~\\ref{thm:primitive digraph}).\nIn Section~\\ref{sec:tournaments}, we take care of it to eventually compute the competition period and competition index of a tournament with sinks (Theorem~\\ref{thm:tournament2}).\n\nFor a positive integer $n$, we denote the set $\\{1,2,\\ldots,n\\}$ by $[n]$ for simplicity.\n\n\\section{Acyclic multipartite tournaments}\\label{sec:acyclic}\n\nIn this section, we compute the competition period and competition index of an acyclic multipartite tournament.\n\n\\begin{Prop}[\\cite{eoh2020m}]\\label{prop:acyclic-digraph}\nFor a digraph $D$, the following are equivalent.\n\\begin{itemize}\n  \\item[(i)] $D$ is acyclic.\n  \\item[(ii)] $W_{\\zeta(D)}=V\\left(D_{\\zeta(D)}\\right)\\neq\\emptyset$.\n  \\item[(iii)] $\\bigcup_{i=0}^{\\zeta(D)}W_i=V(D)$.\n\\end{itemize}\n\\end{Prop}\n\nThe following lemma is a stronger version of Proposition 2.3 given by Eoh {\\it et al.}~\\cite{eoh2020m} in the sense that, regarding the set $\\mathcal{L}$ of lengths of directed walks with an initial vertex in $W_i$, $\\max{\\mathcal{L}} \\le i$ is replaced with $\\mathcal{L}=\\{0,\\ldots,i\\}$.\nAs a matter of fact, their proof asserted this stronger version.\n\n\n\\begin{Lem}[Restatement of Proposition 2.3 in \\cite{eoh2020m}]\\label{lem:walk-length}\nLet $D$ be a digraph with $\\zeta(D)\\ge1$ and $(W_0,\\ldots,W_{\\zeta(D)})$ be the sink sequence of $D$.\nThen, for each $i=0,\\ldots,\\zeta(D)-1$ and each vertex $v$ in $W_i$, among the directed walks starting from $v$, there exist directed walks of lengths $0, \\ldots,i$ and no directed walks of length greater than $i$.\nFurthermore, if $D$ is acyclic, then the statement is true for even $i=\\zeta(D)$.\n\\end{Lem}\n\n\\begin{Thm}\\label{acyclic-digraph-properties}\n  Let $D$ be an acyclic digraph and $\\left(W_0,\\ldots,W_{\\zeta(D)}\\right)$ be its sink sequence.\n   Then, for $\\zeta(D)\\ge1$, we have the following:\n  \\begin{itemize}\n    \\item[(i)] $C^m(D)$ is an empty graph for any integer $m>\\zeta(D)$;\n    \\item[(ii)] {\\rm cperiod}$(D)=1$;\n    \\item[(iii)] if $|W_i|=1$ for some integer $i\\in\\{0,\\ldots,{\\zeta(D)}-1\\}$, then $C^{\\zeta(D)}(D)$ is the union of the complete graph with the vertex set $W_{\\zeta(D)}$ and the empty graph with the vertex set $V(D)\\setminus W_{\\zeta(D)}$;\n    \\item[(iv)] {\\rm cindex}$(D)\\le\\zeta(D)+1$ where the equality holds if $|W_{\\zeta(D)}|>|W_i|$ for some integer $i\\in\\{0,\\ldots,{\\zeta(D)}-1\\}$.\n  \\end{itemize}\n\\end{Thm}\n\\begin{proof}\nTake an integer $m>\\zeta(D)$.\nSince $D$ is acyclic, $\\bigcup_{i=0}^{\\zeta(D)}W_i=V(D)$ by Proposition \\ref{prop:acyclic-digraph} and so no vertex in $D$ has an $m$-step prey in $D$ by Lemma \\ref{lem:walk-length}.\nTherefore $C^m(D)$ is an empty graph and so the statement (i) is true.\nThen, by the definition of competition period, the statement (ii) is immediately true.\n\nTo show the statement (iii), suppose that $|W_i|=1$ for some integer $i\\in\\{0,\\ldots,{\\zeta(D)}-1\\}$.\nLet $W_i=\\{w\\}$.\nSince every vertex in $W_j$ has at least one out-neighbor in $W_{j-1}$ for each integer $1 \\le j \\le \\zeta(D)$, there is a directed walk of length $\\zeta(D)-i$ from $v$ to $w$ for each vertex $v\\in W_{\\zeta(D)}$.\nMoreover, there is a directed walk of length $i$ from $w$ to a vertex $u\\in W_0$.\nBy concatenating those directed walks, we obtain a directed walk of length $\\zeta(D)$ from each vertex in $W_{\\zeta(D)}$ to $u$.\nThus $W_{\\zeta(D)}$ forms a clique in $C^{\\zeta(D)}(D)$.\nBy Proposition~\\ref{prop:acyclic-digraph}(iii), every vertex in $V(D)\\setminus W_{\\zeta(D)}$ belongs to $W_j$ for some $j \\in \\{0, \\ldots, \\zeta(D)-1\\}$.\nTherefore every vertex in $V(D)\\setminus W_{\\zeta(D)}$ is isolated in $C^{\\zeta(D)}(D)$ by Lemma \\ref{lem:walk-length}.\nThus the statement (iii) is true.\n\nThe inequality {\\rm cindex}$(D)\\le\\zeta(D)+1$ immediately follows from (i).\nTo figure out when the equality holds, suppose that $|W_{\\zeta(D)}|>|W_i|$ for some integer $i\\in\\{0,\\ldots,{\\zeta(D)}-1\\}$.\nAs we have observed above, there are at least $|W_{\\zeta(D)}|$ directed walks starting from distinct vertices in $W_{\\zeta(D)}$ to a vertex in $W_{i}$.\nSince $|W_{\\zeta(D)}|>|W_i|$, there are at least\ntwo directed walks terminating at the same vertex in $W_i$ by the pigeonhole principle.\nThen the origins of those directed walks form a clique in $C^{\\zeta(D)}(D)$.\nThus $C^{\\zeta(D)}(D)$ is not an empty graph.\nHence, by the definition of competition index and (i), {\\rm cindex}$(D)=\\zeta(D)+1$.\n\\end{proof}\n\nIt is easy to check that an acyclic multipartite tournament has sink elimination index $\\zeta(D)\\ge1$.\n\n\\begin{Thm}\\label{k-partite tournament if and only if conditon}\n  For an integer $k\\ge2$, let $D$ be an acyclic $k$-partite tournament with a $k$-partition $(V_1,\\ldots,V_k)$. Then $\\left(W_0,\\ldots,W_{\\zeta(D)}\\right)$ is the sink sequence of $D$ if and only if $(W_0,\\ldots,W_{\\zeta(D)})$ is a partition of $V(D)$ satisfying the following:\n  \\begin{itemize}\n   \\item[(i)] for each $i = 0,\\ldots,\\zeta(D)$, $W_i$ is a subset of a partite set of $D$;\n   \\item[(ii)] if there is an arc from a vertex in $W_j$ to a vertex in $W_i$ for some $i,j \\in \\{0,\\ldots, \\zeta(D)\\}$, then $i < j$ and $W_i$ and $W_j$ are included in different partite sets.\n %\n   \\item[(iii)] for each $i = 0,\\ldots,\\zeta(D)-1$,\n   there is an arc from each vertex in $W_{i+1}$ to each vertex in $W_i$.\n  \\end{itemize}\n\\end{Thm}\n\\begin{proof} Suppose that $\\left(W_0,\\ldots,W_{\\zeta(D)}\\right)$ is the sink sequence of $D$.\nLet $(D_0,\\ldots,D_{\\zeta(D)})$ be the digraph sequence associated with it.\nSuppose, to the contrary, that there are two vertices $u,v\\in W_i$ for some $i\\in\\{0,\\ldots,\\zeta(D)\\}$ such that $u\\in V_p$ and $v\\in V_q$ with $p\\neq q$.\nSince $D$ is a $k$-partite tournament, $(u,v)$ or $(v,u)$ is an arc in $D$.\nWithout loss of generality, we may assume that $(u,v)$ is an arc in $D$.\nBy the definition of $D_i$, $(u,v)$ is an arc in $D_i$, which contradicts the assumption that $u\\in W_i$.\nThus, for each integer $0 \\le i \\le \\zeta(D)$, $W_i\\subseteq V_s$ for some $s\\in[k]$ and so the statement (i) is true.\n\nSuppose there is an arc from a vertex $W_j$ to a vertex $W_i$ for some $i,j\\in\\{0,\\ldots,\\zeta(D)\\}$.\nThen, by the definition of sink sequence, $i<j$.\nBy (i), $W_i \\subseteq V_p$ and $W_j \\subseteq V_q$ for some $p, q \\in [k]$.\nSince an arc goes from $W_j$ to $W_i$, $p \\neq q$ and so the statement (ii) is true.\n\nBy definition, $W_i$ and $W_{i+1}$ are not empty sets for each $i = 0,\\ldots,\\zeta(D)-1$, so there exists an arc from a vertex in $W_{i+1}$ to a vertex in $W_i$.\nTherefore by (ii), $W_i$ and $W_{i+1}$ are included in different partite sets.\nSince $D$ is a multipartite tournament, there is an arc from each vertex in $W_{i+1}$ to each vertex in $W_i$.\nHence (iii) is true and so the `only if' part is valid.\n\nConversely, consider a partition $(W_0,\\ldots,W_{\\zeta(D)})$ of $V(D)$ satisfying (i), (ii), and (iii).\nBy (ii), every vertex in $W_0$ is a sink of $D$. Suppose that there is a sink $v$ of $D$ in $V(D)\\setminus W_0$.\nThen, since $\\bigcup_{j=0}^{\\zeta(D)}W_j = V(D)$, $v\\in W_i$ for some integer $i\\in[\\zeta(D)]$.\nBy (iii), $W_i$ and $W_{i-1}$ are included in distinct partite sets. Since $D$ is a multipartite tournament and (ii) is true, there is an arc from $v$ to a vertex in $W_{i-1}$, which contradicts the choice of $v$.\nTherefore $W_0$ is the set of sinks in $D$.\nBy applying a similar argument to $W_1$ of the subdigraph $D_1$ induced by $V(D)\\setminus W_0$, we may show that $W_1$ is the set of sinks of $D_1$.\nInductively, we may show that $W_i$ is the set of sinks of the subdigraph of $D$ induced by $V(D) \\setminus \\bigcup_{j=0}^{i-1}W_j$ for $i=2,\\ldots,\\zeta(D)$.\nSince $(W_0,\\ldots,W_{\\zeta(D)})$ is a partition of $V(D)$, we may conclude that $(W_0,\\ldots,W_{\\zeta(D)})$ is the sink sequence of $D$ and so the `if' part is true.\n\\end{proof}\n\n\\begin{Cor}\\label{cor:zeta condition}\nFor an integer $k\\ge2$, let $D$ be an acyclic $k$-partite tournament with a $k$-partition $(V_1,\\ldots,V_k)$. If $\\left(W_0,\\ldots,W_{\\zeta(D)}\\right)$ is the sink sequence of $D$, then\n$\\zeta(D)\\ge k-1$ where the equality holds if and only if $W_i$ is a partite set of $D$ for each $i=0,\\ldots,\\zeta(D)$;\n\\end{Cor}\n\\begin{proof} Since $D$ is acyclic, $\\bigcup_{i=0}^{\\zeta(D)}W_i=V(D)$ by Proposition~\\ref{prop:acyclic-digraph}.\n By Theorem~\\ref{k-partite tournament if and only if conditon}(i), $W_i$ is a subset of a partite set of $D$ for each integer $0 \\le i \\le \\zeta(D)$. If $\\zeta(D)<k-1$, then there exists a partite set not containing $W_i$ for any $i\\in\\{0,\\ldots,\\zeta(D)\\}$, which is impossible. Thus $\\zeta(D)\\ge k-1$. It is obvious that $\\zeta(D)=k-1$ if and only if $\\{W_0,\\ldots,W_{\\zeta(D)}\\}=\\{V_1,\\ldots,V_k\\}$.\n\\end{proof}\nIf $D$ is an acyclic $k$-partite tournament for an integer $k\\geq 3$, then $\\zeta(D) \\ge 2$ by Corollary~\\ref{cor:zeta condition} and we have the following result.\n\\begin{Cor}\\label{lem:three are different}\n Let $D$ be an acyclic $k$-partite tournament for an integer $k\\ge3$ and let $\\left(W_0,\\ldots,W_{\\zeta(D)}\\right)$ be the sink sequence of $D$.\n   Then there is an integer $i\\in\\{0,\\ldots,\\zeta(D)-2\\}$ such that $W_{i}$ and $W_{i+2}$ are included in different partite sets.\n\\end{Cor}\n\\begin{proof}\nLet $(V_1,\\ldots, V_k)$ be a $k$-partition of $D$.\nBy Theorem~\\ref{k-partite tournament if and only if conditon}(iii), $W_{0}$ and $W_{1}$ are included in different partite sets.\nWithout loss of generality, we may assume that $W_{0}\\subseteq V_1$ and $W_{1}\\subseteq V_2$.\nSuppose, to the contrary, that $W_{i}$ and $W_{i+2}$ are included in the same partite set for each integer $0 \\le i \\le \\zeta(D)-2$.\nThen\n\\begin{equation*}\n\\bigcup_{0 \\le i \\le {\\zeta(D)}\/2}{W_{2i}} \\subseteq V_1 \\quad \\mbox{ and } \\quad \\bigcup_{0 \\le i \\le ({\\zeta(D)}-1)\/2}{W_{2i+1}} \\subseteq V_2.\n\\end{equation*}\nTherefore $\\bigcup_{i=0}^{\\zeta(D)}W_i\\subseteq V_1\\cup V_2$.\nSince $k\\ge 3$, $V_3\\neq \\emptyset$.\nThus $\\bigcup_{i=0}^{\\zeta(D)}W_i\\subsetneq V_1\\cup V_2\\cup V_3\\subseteq V(D)$, which contradicts Proposition~\\ref{prop:acyclic-digraph}(iii).\n\\end{proof}\n\n\\begin{Lem}\\label{Lem:directed path}\n  Let $D$ be an acyclic $k$-partite tournament for an integer $k\\ge3$ and let $\\left(W_0,\\ldots,W_{\\zeta(D)}\\right)$ be the sink sequence of $D$.\n  For each integer $1\\le s \\le \\zeta(D)$, if there exist integers $p, q \\in \\{0,\\ldots,s\\}$ with $p>q$ such that $W_{p}$ and $W_{q}$ are included in different partite sets, then there exists a directed path of length $s-p+q+1$ from each vertex in $W_s$ to each vertex in $W_0$.\n\\end{Lem}\n\n\\begin{proof}\nFix $s\\in \\{1,\\ldots,\\zeta(D)\\}$.\nSuppose that there exist integers $p, q \\in \\{0,\\ldots,s\\}$ with $p>q$ such that $W_{p}$ and $W_{q}$ are included in different partite sets.\nTake vertices $u\\in W_{s}$ and $x\\in W_0$.\nNow we take two vertices $v\\in W_{p}$ and $w \\in W_q$ so that $u=v$ if $p=s$ and $w=x$ if $q=0$.\n  Since $D$ is a multipartite tournament, every vertex in $W_{i-1}$ is an out-neighbor of each vertex in $W_i$ for each integer  $1\\le i \\le \\zeta(D)$.\nTherefore, there exist a $(u,v)$-directed path $P$ of length $s-p$ and a $(w,x)$-directed path $Q$ of length $q$.\nSince $W_{p}$ and $W_{q}$ are included in different partite sets, $v$ and $w$ are linked by an arc.\nThen, by Theorem~\\ref{k-partite tournament if and only if conditon}(ii), $(v,w)$ is an arc of $D$.\nThus $P\\to Q$ is a $(u,x)$-directed path of length $(s-p)+1+q$.\nSince $u$ and $x$ were arbitrarily chosen, the statement is true.\n\\end{proof}\n\nWe recall that if $D$ is an acyclic $k$-partite tournament for an integer $k\\geq 3$, then $\\zeta(D) \\ge 2$.\n\n\\begin{Thm}\\label{Thm:acyclic k-partite tournament}\n  For an integer $k\\geq 3$, let $D$ be an acyclic $k$-partite tournament with the sink sequence $\\left(W_0,\\ldots ,W_{\\zeta(D) }\\right)$.\n  Then the following are true:\n  \\begin{itemize}\n    \\item[(i)] $C^{\\zeta(D)}(D)$ is the union of the complete graph with the vertex set $W_{\\zeta(D)}$ and the empty graph with the vertex set $V(D)\\setminus W_{\\zeta(D)}$;\n          \\item[(ii)] $C^{\\zeta(D)-1}(D)$ is the union of the complete graph with the vertex set $W_{\\zeta(D)}\\cup W_{\\zeta(D)-1}$ and the empty graph with the vertex set $\\bigcup _{i=0}^{\\zeta(D)-2}W_{i}$;\n  \n    \\item[(iii)] if $\\left\\vert W_{\\zeta(D)}\\right\\vert \\geq 2$, then ${\\rm cindex}(D)=\\zeta(D)+1$, otherwise ${\\rm cindex}(D)=\\zeta(D)$.\n  \\end{itemize}\n\\end{Thm}\n\\begin{proof}\nTake a vertex $z \\in W_0$.\nBy Theorem~\\ref{k-partite tournament if and only if conditon}(iii),\n$W_0$ and $W_1$ are included in different partite sets.\nThen, since $0 < 1 \\le \\zeta(D)-1$, by Lemma~\\ref{Lem:directed path}, there exist\n\\begin{itemize}\n  \\item[(a)] a directed path of length $\\zeta(D)$ from any vertex in $W_{\\zeta(D)}$ to $z$ and\n  \\item[(b)] a directed path of length $\\zeta(D)-1$ from any vertex in $W_{\\zeta(D)-1}$ to $z$.\n\\end{itemize}\nBy (a), $W_{\\zeta(D)}$ forms a clique in $C^{\\zeta(D)}(D)$.\nBy Lemma \\ref{lem:walk-length}, every vertex in $\\bigcup _{i=0}^{\\zeta(D)-1}W_{i}$ is isolated in $C^{\\zeta(D)}(D)$.\nBy Proposition~\\ref{prop:acyclic-digraph}, $\\bigcup _{i=0}^{\\zeta(D)-1}W_{i} = V(D)\\setminus W_{\\zeta(D)}$ and so the statement (i) is true.\n\nSince $D$ is a $k$-partite tournament with $k \\ge 3$, there is an integer $i \\in \\{0,\\ldots, \\zeta(D)-2\\}$ such that  $W_{i}$ and $W_{i+2}$ are included in different partite sets by Corollary \\ref{lem:three are different}.\nThus, by Lemma~\\ref{Lem:directed path}, there is a directed path of length $\\zeta(D)-1$ from each vertex in $W_{\\zeta(D)}$ to $z$.\nHence, by (b), $z$ is a $(\\zeta(D)-1)$-step common prey of each vertex in $W_{\\zeta(D)} \\cup W_{\\zeta(D)-1}$ and so $W_{\\zeta(D)} \\cup W_{\\zeta(D)-1}$ forms a clique in $C^{\\zeta(D)-1}(D)$.\nBy Lemma \\ref{lem:walk-length}, every vertex in $\\bigcup _{i=0}^{\\zeta(D)-2}W_{i}$ is isolated in $C^{\\zeta(D)-1}(D)$ and so, by Proposition~\\ref{prop:acyclic-digraph}, the statement (ii) is true.\n\n\nSuppose $\\left\\vert W_{\\zeta(D)}\\right\\vert \\geq 2$.\nThen, by the statement (i), $C^{\\zeta(D)}(D)$ is not empty.\nSince $C^m(D)$ is empty for each integer $m>\\zeta(D)$ by Theorem~\\ref{acyclic-digraph-properties}(i), ${\\rm cindex}(D)$ is $\\zeta(D)+1$.\nNow suppose $\\left\\vert W_{\\zeta(D)}\\right\\vert \\le 1$.\nThen, since $D$ is acyclic, $\\left\\vert W_{\\zeta(D)}\\right\\vert = 1$\nand so, by the statements (i) and (ii), $C^{\\zeta(D)}(D)$ is empty and $C^{\\zeta(D)-1}(D)$ is not empty, respectively.\nSince $C^m(D)$ is empty for each integer $m>\\zeta(D)$ by Theorem~\\ref{acyclic-digraph-properties}(i), ${\\rm cindex}(D)$ is $\\zeta(D)$.\nTherefore the statement (iii) is true.\n\\end{proof}\n\n\n\\section{Multipartite tournaments with sinks and directed cycles}\\label{sec:directed cycle}\n\nIf a digraph $D$ is acyclic, then $C^m(D)$ is empty for any integer $m > \\zeta(D)$ and so the competition period of $D$ is $1$ (Theorem~\\ref{acyclic-digraph-properties}).\nIn addition, Cho and Kim~\\cite{cho2004competition} showed that a digraph without sinks has competition period $1$.\nIn this vein, this section studies the competition period of a multipartite tournament having a sink and a directed cycle.\nBy the way, Eoh {\\it et al.}~\\cite{eoh2020m} showed that if $C^M(D)$ is an empty graph for a digraph $D$ and a positive integer $M$, then so is $C^m(D)$ for any positive integer $m \\ge M$.\nTherefore, if $C^M(D)$ is an empty graph for a digraph $D$ and a positive integer $M$, then the competition period of $D$ is $1$.\nAs a matter of fact, in the case of a multipartite tournament $D$, having a sink and a directed cycle guarantees that $C^m(D)$ is not empty for every positive integer $m$ by the following proposition.\n\n\\begin{Prop}\\label{Prop:notempty}\n  If a multipartite tournament $D$ has a sink and a directed cycle, then $C^m(D)$ is not empty for every positive integer $m$.\n\\end{Prop}\n\n\\begin{proof}\n    Suppose that a $k$-partite tournament $D$ for an integer $k \\ge 2$ has a sink $x$ and a directed cycle $C:=v_0v_1 \\cdots v_{l-1} v_0$ for some integer $l \\ge 3$.\n    Let $(V_1,\\ldots,V_k)$ be a $k$-partition of $D$.\n  Without loss of generality, we may assume that $x \\in   V_1$.\n  Since $V_1$ is a partite set of $D$, there exist at least two vertices of $V(C) \\setminus V_1$.\n  Let $v_i$ and $v_j$ be vertices of $V(C)\\setminus V_1$ for some two distinct integers $i,j \\in \\{0,1,\\ldots,l-1\\}$.\n  Since $D$ is a $k$-partite tournament and $x$ is a sink, $(v_i,x)$ and $(v_j,x)$ are arcs of $D$.\n  Then $v_i$ and $v_j$ are adjacent in $C(D)$.\n  Since $v_i$ and $v_j$ are on $C$, $v_{i-m+1}$ and $v_{j-m+1}$ are adjacent in $C^m(D)$ for every positive integer $m$ (all the subscripts are reduced to modulo $l$).\n  Hence $C^m(D)$ is not empty for every positive integer $m$.\n\\end{proof}\n\nIn the following, we shall show that if a multipartite tournament $D$ has a sink and a directed cycle, then the competition period of $D$ is at most three, which is our main result.\nTo do so, we need several theorems and lemmas.\n\n\\begin{Thm}[\\cite{goddard1991multipartite}]\\label{thm:goddard1991multipartite}\n  Let $D$ be a $k$-partite tournament with $k\\ge 3$.\n  Then $D$ contains a directed cycle of length $3$ if and only if there exists a directed cycle in $D$ which contains vertices from at least three partite sets.\n\\end{Thm}\n\nGiven a digraph $D$, we call a directed cycle of length at least $4$ in $D$, a \\textit{directed hole} if it is an induced subdigraph of $D$.\n\nWe note that a directed hole of length $4$ in a multipartite tournament $D$ is of the form $v_0 \\to v_1 \\to v_2 \\to v_3 \\to v_0$ such that $\\{v_0,v_2\\} \\subseteq X$ and $\\{v_1,v_3\\} \\subseteq Y$ for some distinct partite sets $X$ and $Y$ of $D$.\n\nGiven a multipartite tournament $D$, if a directed walk contains vertices which induce a directed cycle of length $3$ in $D$ (resp.\\ a directed hole of length $4$ in $D$), then we say that it is of \\emph{Type~1} (resp.\\ \\emph{Type~2}).\n\n\\begin{Lem}\\label{lem:34cycle1}\nLet $D$ be a multipartite tournament of order $n \\ge 3$.\n  Then any directed walk of length at least $n$ in $D$ is of Type~1 or Type~2.\n\\end{Lem}\n\n\\begin{proof}\n  Let $Q$ be a directed walk of length at least $n$ in $D$.\n  Since the length of $Q$ is greater than or equal to the number of vertices of $D$, $Q$ contains a directed cycle $C:= u_0 \\to u_1 \\to u_2 \\to \\cdots \\to u_{l-1} \\to u_0$ for an integer $l \\ge 3$.\n\n  {\\it Case 1.} There are at least $3$ vertices on $C$ which belong to distinct partite sets in $D$.\nWe may apply Theorem~\\ref{thm:goddard1991multipartite} to the multipartite tournament induced by $V(C)$ to conclude that there is a directed cycle of length $3$ all of whose vertices are on $C$.\nThen it is easy to check that $Q$ is of Type~1.\n\n  {\\it Case 2.}  There exist two distinct partite sets $X$ and $Y$ in $D$ such that $V(C) \\subseteq X \\cup Y$.\n  Then $l$ is even, so $l \\ge 4$.\n  Without loss of generality, we may assume that a vertex on $C$ with an even index belongs to $X$ and a vertex on $C$ with an odd index belongs to $Y$.\n  Since $D$ is a $k$-partite tournament, there is an arc between $u_i$ and $u_{l-i-1}$ for each integer $0 \\le i \\le l\/2 - 1$.\nSince $(u_{l-1},u_0) \\in A(D)$ and $C$ is a directed cycle, $(u_{i},u_{l-i-1})$ is an arc in $D$ for some $i \\in \\{1,\\ldots, l\/2 -1\\}$.\nWe may regard $i$ as the smallest index among $1, \\ldots, l\/2 -1$ such that $(u_{i},u_{l-i-1})$ is an arc in $D$.\n  Then $(u_{l-i},u_{i-1}) \\in A(D)$ and $C' := u_{i-1} \\to u_i \\to u_{l-i-1} \\to u_{l-i} \\to u_{i-1}$ is a directed hole of length $4$ in $D$.\n  Thus $Q$ is of Type~2.\n \\end{proof}\n\n\nWe note that a multipartite tournament has a sink if and only if the sink elimination index is greater than or equal to one.\nIn the following five results prior to our main result, we examine lengths of directed walks from a vertex in $D_{\\zeta(D)}$ to a vertex in $\\bigcup_{I=0}^{\\zeta(D)-1}W_i$ where $(W_0,\\ldots,W_{\\zeta(D)})$ is the sink sequence of a multipartite tournament $D$ with a directed cycle and $\\zeta(D) \\ge 1$.\n\n\n\n\\begin{Lem}\\label{lem:3cycle}\n Let $D$ be a multipartite tournament.\n  Suppose that there exists a $(u,v)$-directed walk of Type~1 of length $\\ell$ for some vertices $u$ and $v$.\nThen there is a $(u,v)$-directed walk of length $\\ell+3m$ for each nonnegative integer $m$.\n\\end{Lem}\n\n\\begin{proof}\n  Let $Z$ be a $(u,v)$-directed walk of length $\\ell$ that contains three vertices which induce a directed cycle $C$ of length $3$ in $D$.\n  Let $x$ be a vertex on $C$, $Z_1$ be a $(u,x)$-section of $Z$, and $Z_2$ be the $(x,v)$-section of $Z$ obtained by cutting $Z_1$ away from $Z$.\n  We may assume that the sequence representing $C$ starts at $x$.\n  Then, for a nonnegative integer $m$, we may create a $(u,v)$-directed walk of length $\\ell + 3m$ in such a way that we traverse $Z_1$, $C$ as many as $m$ times, and then $Z_2$.\n\\end{proof}\n\n\n\n\\begin{Lem}\\label{lem:4cycle}\nLet $D$ be a multipartite tournament\nwith the sink elimination index $\\zeta(D) \\ge 1$ and\nthe sink sequence $\\left(W_0,\\ldots,W_{\\zeta(D)}\\right)$.\n  For a vertex $u$, suppose that there exists a $(u,w)$-directed walk $Z$ of Type~2 of length $\\ell$ for some vertex $w$ such that (i) $w$ belongs to $\\bigcup_{i=0}^{\\zeta(D)-1}W_i$ or\n  (ii) there exists a directed hole of length $4$ such that its four vertices are on $Z$ and $w$ does not belong to any of two partite sets which the four vertices belong to.\nThen there exists a positive integer $N$ such that for each integer $m \\ge N$, there is a $(u,w)$-directed walk of length $\\ell+2m$.\n\\end{Lem}\n\n\\begin{proof}\nSince $Z$ is of Type~2, there are four vertices on $Z$ which induce a directed hole $H:=v_0 \\to v_1 \\to v_2 \\to v_3 \\to v_0$ of length $4$ in $D$.\n Then there exist two distinct partite sets $X$ and $Y$ of $D$ such that $\\{v_0,v_2\\} \\subseteq X$ and $\\{v_1,v_3\\} \\subseteq Y$.\nIf the case (ii) of the lemma statement happens, we may assume that $H$ is the hole mentioned in the case.\nLet $Z_1$ be a $(u,v_0)$-section of $Z$, and $Z_2$ be the $(v_0,w)$-section of $Z$ obtained by cutting $Z_1$ away from $Z$.\n   We denote the lengths of $Z_1$ and $Z_2$ by $\\ell_1$ and $\\ell_2$, respectively.\n\n  {\\it Case 1.} There exists a vertex $z$ on $Z_2$ which does not belong to $X \\cup Y$.\n  Let $Z_3$ be a $(v_0,z)$-section of $Z_2$, and $Z_4$ be the $(z,w)$-section of $Z_2$ obtained by cutting $Z_3$ away from $Z_2$.\n   We denote the lengths of $Z_3$ and $Z_4$ by $\\ell_3$ and $\\ell_4$, respectively.\n  We consider the two subcases: $(v_i,z) \\in A(D)$ for each $i =0,1,2,3$; there is an arc $(z,v)$ for some vertex $v$ on $H$.\n  Suppose $(v_i,z) \\in A(D)$ for each $i =0,1,2,3$ and fix a positive integer $\\alpha$.\n   Then, traverse the directed walk $Z_1$, go around $H$ as many times as desired, depart it at an appropriate vertex on $H$ to reach $z$ by an arc, and traverse $Z_4$.\n   This creates a $(u,w)$-directed walk of length $\\ell_1 + \\alpha + \\ell_4$.\n\n  Now suppose that there is an arc $(z,v)$ for some vertex $v$ on $H$.\n  Let $D^*$ be the subdigraph of $D$ induced by $V(Z_3) \\cup V(H)$.\n  Then $D^*$ contains vertices from at least three partite sets.\n  Since $D^*$ is a multipartite tournament,\n    $D^*$ contains a directed cycle $C$ of length $3$ by Theorem~\\ref{thm:goddard1991multipartite}.\n  Moreover, the directed walk $Z_3$, the arc $(z,v)$, and $(v,v_0)$-section of $H$ form a closed directed walk $Q$ in $D^*$.\n    Then $V(Q) \\cup V(H) = V(D^*)$ and $V(Q) \\cap V(H) \\neq \\emptyset$, so $D^*$ is strongly connected.\n    Since $D^*$ contains the directed cycles $C$ of length $3$ and $H$ of length $4$, we may conclude that $D^*$ is primitive.\n    Thus, for any $\\beta \\ge \\exp(D^*)$, there is a $(v_0,z)$-directed walk of length $\\beta$ and so there is a $(u,w)$-directed walk of length $\\ell_1 + \\beta + \\ell_4$.\n\n    Since $\\ell_1 + \\ell_4 \\le \\ell$ and $\\alpha$ and $\\beta$ were arbitrarily chosen among the positive integers bounded below, we have shown in both subcases that there exists a sufficiently large $N$ such that there is a $(u,w)$-directed walk of length $\\ell+2m$ for each integer $m \\ge N$.\n\n  {\\it Case 2.} Every vertex on $Z_2$ belongs to $X \\cup Y$.\n  Then the case (i) of the lemma statement happens, that is, $w \\in \\bigcup_{i=0}^{\\zeta(D)-1}W_i$.\n  Let $j=0$ if $w \\in Y$ and $j=1$ if $w \\in X$.\n  Then $2m+j+1$ and $\\ell_2$ have the same parity.\n  For, $\\ell_2$ is odd if $w \\in Y$ and $\\ell_2$ is even if $w \\in X$ since $v_0 \\in X$ and $Z_2$ is a $(v_0,w)$-directed walk whose vertices belong to $X \\cup Y$.\n  Furthermore, $w$ and any of $v_j, v_{j+2}$ belong to distinct partite sets.\n  Since $w\\in\\bigcup_{i=0}^{\\zeta(D)-1}W_i$ and $\\{v_0,v_1,v_2,v_3\\}\\subseteq V(D_{\\zeta(D)})$, there are arcs $(v_j,w)$ and $(v_{j+2},w)$ in $D$.\n     Now let\n     \\[\n     \\Theta_i = Z_1 \\to H_i \\to H' \\to w\n     \\]\n     %\n     where $H'$ denotes the $(v_0,v_j)$-section of $H$ and $H_i$ means the directed walk obtained by going around $H$ $i$ times, for a nonnegative integer $i$.\n     Then $\\Theta_i$ is a $(u,w)$-directed walk in $D$.\n     For a nonnegative integer $i$, we denote by $\\Lambda_i$ the directed walk obtained from $\\Theta_i$ by replacing the arc $(v_j,w)$ with the directed path $v_j \\to v_{j+1} \\to v_{j+2} \\to w$.\n    Then the lengths of $\\Theta_i$ and $\\Lambda_i$ are $\\ell_1 + 4i + j+1$ and $\\ell_1 + 4i + j+3$, respectively, for each nonnegative integer $i$.\n    Accordingly, we have shown that there is a $(u,w)$-directed walk of length $\\ell_1 + 2m+j+1$ for each nonnegative integer $m$.\n    Since $2m+j+1$ and $\\ell_2$ have the same parity, we have actually shown that there exists $(u,w)$-directed walk of length $\\ell + 2m$ for each nonnegative integer $m$.\n  %\n\n\\end{proof}\n\nLet $p_1,\\ldots, p_t$ be positive integers with\n$\\gcd(p_1,\\ldots,p_t)=1$.\nThe {\\em Frobenius number} of\n$p_1,\\ldots, p_t$ is the largest integer $b$ for which the\nFrobenius equation $$p_1x_1+\\cdots+p_tx_t=b$$ has no nonnegative\ninteger solution $(x_1,\\ldots,x_t)$.\nThe number $b$ is denoted by $F(p_1,\\ldots,p_t)$.\n\n\\begin{Lem}\\label{lem:frobenius}\nLet $D$ be a multipartite tournament\nwith the sink elimination index $\\zeta(D) \\ge 1$ and\nthe sink sequence $\\left(W_0,\\ldots,W_{\\zeta(D)}\\right)$.\n  For a vertex $u \\in V(D_{\\zeta(D)})$, suppose that there exists a $(u,w)$-directed walk of length $\\ell$ for some vertex $w\\in \\bigcup_{i=0}^{\\zeta(D)-1}W_i$ such that its sequence contains both a directed cycle of length $3$ and a directed hole of length $4$.\nThen there exists a positive integer $N$ such that for each integer $m \\ge N$, there is a $(u,w)$-directed walk of length $\\ell+m$.\n  \\end{Lem}\n\n\\begin{proof}\n  Let $Z$ be a $(u,w)$-directed walk of length $\\ell$ whose sequence contains a directed cycle $C$ of length $3$ and a directed hole $H$ of length $4$.\n  If (i) the sequence of $C$ appears before that of $H$ on $Z$, then there exists a vertex on $H$ such that, on $Z$, all the vertices on $C$ appear before it and all the vertices on $H$ appear after it.\n  We denote such a vertex by $x$.\n  Suppose that (ii) the sequence of $H$ appears before that of $C$ on $Z$.\n    Since the vertices on $H$ belong to two distinct partite sets and the vertices on $C$ belong to three distinct partite sets, there exists a vertex $y$ on $C$ which does not belong to any of two partite sets which contain the vertices on $H$.\n\n  Now we denote by $v$ the vertex $x$ if (i) happens, and the vertex $y$ if (ii) occurs.\n  Let $Z_1$ be a $(u,v)$-section of $Z$, and $Z_2$ be the $(v,w)$-section of $Z$ obtained by cutting $Z_1$ away from $Z$.\n   We denote the lengths of $Z_1$ and $Z_2$ by $\\ell_1$ and $\\ell_2$, respectively.\n    Fix a nonnegative integer $t$.\n     Since $F(2,3)=1$, there exist nonnegative integers $m_1, m_2$ such that $t = 3m_1 + 2m_2 -2 $.\n\n     Consider the case (i).\n     Then $v=x$.\n     Since $Z_1$ is a $(u,v)$-directed walk of Type~1, by Lemma~\\ref{lem:3cycle}, there exists a $(u,v)$-directed walk $Q_1$ of length $\\ell_1 + 3m_1$.\n   Since $Z_2$ is a $(v,w)$-directed walk of Type~2, by Lemma~\\ref{lem:4cycle}, there exists a positive integer $N'$ such that there is a $(v,w)$-directed walk $Q_2$ of length $\\ell_2+2(m_2 + N')$.\n   Then $Q_1 \\to Q_2$ is a $(u,w)$-directed walk of length\n   \\[\n   (\\ell_1 + 3m_1) + (\\ell_2 + 2(m_2+N')) = \\ell + 2N' +2 + (3m_1 + 2m_2 -2)= \\ell + 2N'+2 + t.\n   \\]\n\n    Consider the case (ii).\n    Then $v=y$.\n   By Lemma~\\ref{lem:4cycle}, there exists a positive integer $N''$ such that there is a $(u,v)$-directed walk $Q'_1$ of length $\\ell_1+2(m_2 + N'')$.\n    By applying Lemma~\\ref{lem:3cycle} to the $(v,w)$-directed walk $C \\to Z_2$, there exists a $(v,w)$-directed walk $Q'_2$ of length $3+ \\ell_2 + 3m_1$.\n   Then $Q'_1 \\to Q'_2$ is a $(u,w)$-directed walk of length\n   \\[\n    (\\ell_1 + 2(m_2+N'')) + (3+ \\ell_2 + 3m_1)   = \\ell + 2N'' +5 + (3m_1 + 2m_2 -2)= \\ell + 2N''+5 + t.\n   \\]\n    Let\n   \\[\n   N=\n   \\begin{cases}\n     2N'+2, & \\mbox{if } v=x; \\\\\n     2N''+5, & \\mbox{if } v=y.\n   \\end{cases}\n   \\]\n   Then, since $t$ was chosen as an arbitrary nonnegative integer, we have shown that, for each integer $m \\ge N$, there is a $(u,w)$-directed walk of length $\\ell + m$.\n\\end{proof}\n\n\n \\begin{Lem}\\label{lem:both34}\n Let $D$ be a multipartite tournament\nwith a directed cycle, the sink elimination index $\\zeta(D) \\ge 1$, and\nthe sink sequence $\\left(W_0,\\ldots,W_{\\zeta(D)}\\right)$.\n  For a vertex $u \\in V(D_{\\zeta(D)})$, suppose that there exist a $(u,w_1)$-directed walk $Q_1$ of Type~1 and a $(u,w_2)$-directed walk $Q_2$ of Type~2 for some vertices $w_1,w_2\\in\\bigcup_{i=0}^{\\zeta(D)-1}W_i$.\n  Then there exists a positive integer $N$ such that\n   there is a $(u,w)$-directed walk of length $\\ell$ for each vertex $w \\in  \\bigcup_{i=0}^{\\zeta(D)-1}W_i$ and any integer $\\ell \\ge N$.\n\\end{Lem}\n\n\\begin{proof}\nTake $w \\in  \\bigcup_{i=0}^{\\zeta(D)-1}W_i$.\nLet $C$ be a directed cycle of length $3$ in $D$ induced by three vertices in $Q_1$ and\n $H$ be a directed hole of length $4$ in $D$ induced by four vertices in $Q_2$.\n Then there are three distinct partite sets of $D$ to which the vertices on $C$ belong.\n  Since the vertices on $H$ belong to two distinct partite sets, there must be an arc linking a vertex $x$ on $C$ and a vertex $y$ on $H$.\n  Moreover, there exist an arc from a vertex  $y'$ on $H$ to $w$ and an arc from a vertex $x'$ on $C$ to $w$ since $w \\in  \\bigcup_{i=0}^{\\zeta(D)-1}W_i$ and the vertices on directed cycles belong to $V(D_{\\zeta(D)})$.\n\nIf $(x,y) \\in A(D)$ (resp.\\ $(y,x) \\in A(D)$), then a $(u,x)$-section of $Q_1$, $C$ starting at $x$, the arc $(x,y)$, $H$ starting at $y$, the $(y,y')$-section of $H$, and the arc $(y',w)$ (resp.\\ a $(u,y)$-section of $Q_2$, $H$ starting at $y$, the arc $(y,x)$, $C$ starting at $x$, the $(x,x')$-section of $C$, and the arc $(x',w)$) form a $(u,w)$-directed walk $Z_w$ whose sequence contains both a directed cycle of length $3$ and a directed hole of length $4$.\nThen there exists a positive integer $N_w$ such that, for each integer $m \\ge N_w$, there is a $(u,w)$-directed walk of length $\\ell_w+m$ by Lemma~\\ref{lem:frobenius} where $\\ell_w$ is the length of $Z_w$.\nIf we denote by $N$ the maximum of such integers $\\ell_w + N_w$, there exists a $(u,w)$-directed walk of length $\\ell$ for each vertex $w \\in  \\bigcup_{i=0}^{\\zeta(D)-1}W_i$ and each integer $\\ell \\ge N$.\n\\end{proof}\n\n\\begin{Thm}\\label{thm:34cycle}\n    Let $D$ be a multipartite tournament\nwith a directed cycle, the sink elimination index $\\zeta(D) \\ge 1$, and\nthe sink sequence $\\left(W_0,\\ldots,W_{\\zeta(D)}\\right)$.\n    Then, for each vertex $w\\in \\bigcup_{i=0}^{\\zeta(D)-1}W_i$, one of the following properties is true:\n  \\begin{itemize}\n  \\item[(i)] for every vertex $u$ in $D_{\\zeta(D)}$, there is a $(u,w)$-directed walk of Type~1;\n    \\item[(ii)] for every vertex $u$ in $D_{\\zeta(D)}$, there is a $(u,w)$-directed walk of Type~2.\n  \\end{itemize}\n\\end{Thm}\n\n\\begin{proof}\n Suppose that $D$ has $n$ vertices and fix vertex $w \\in \\bigcup_{i=0}^{\\zeta(D)-1}W_i$.\n Since $D$ has a directed cycle, $n \\ge 3$.\n We fix $u \\in V(D_{\\zeta(D)})$.\n  Since $D_{\\zeta(D)}$ has no sinks, there is a directed walk $Q_u$ in $D_{\\zeta(D)}$ of length at least $n$ starting at $u$.\n  Since $D_{\\zeta(D)}$ is a subdigraph of $D$, $Q_u$ is a directed walk in $D$.\n  By Lemma~\\ref{lem:34cycle1}, $Q_u$ is of Type~1 or Type~2.\n  Then $Q_u$ contains vertices which induce a directed cycle $C_u$ in $D$ which has a length $3$ or is a directed hole of length $4$.\n  Since the vertices on $C_u$ belong to at least two distinct partite sets, there must be a vertex $y_u$ on $C_u$ which belongs to a partite set distinct from the one to which $w$ belongs.\n  Since $D$ is a multipartite tournament, there must be an arc linking $y_u$ and $w$.\n  Yet, $w\\in \\bigcup_{i=0}^{\\zeta(D)-1}W_i$ implies $(y_u,w) \\in A(D)$.\n  Then the $(u,y_u)$-section of $Q_u$, $C_u$ starting at $y_u$, and the arc $(y_u,w)$ form a $(u,w)$-directed walk of Type~1 or Type~2 in $D$.\n   Thus we have shown that\n  \\begin{itemize}\n    \\item[($\\star$)]  for every vertex $u$ in $D_{\\zeta(D)}$, there is a $(u,w)$-directed walk of Type~1 or Type~2.\n  \\end{itemize}\n\n  For every vertex $u$ in $D_{\\zeta(D)}$, if there exist a $(u,w)$-directed walk of Type~1 and a $(u,w)$-directed walk of Type~2, then the theorem statement is immediately true.\n\n  Now suppose that there exists a vertex $v$ in $D_{\\zeta(D)}$ such that either there is no $(v,w)$-directed walk of Type~1 or there is no $(v,w)$-directed walk of Type~2.\n  Then, by Lemma~\\ref{lem:34cycle1}, every $(v,w)$-directed walk of length at least $n$ is of Type~2 or every $(v,w)$-directed walk of length at least $n$ is of Type~1.\n  We assume the latter.\n  Then, by $(\\star)$, there exists a $(v,w)$-directed walk $Z_v$ of Type~1.\n   Then $Z_v$ contains vertices which induce a directed cycle $C$ of length $3$ in $D$.\n \n  In the following, we will show that for every vertex $u$ in $D_{\\zeta(D)}$, there is a $(u,w)$-directed walk of Type~1.\n  Take a vertex $u$ in $D_{\\zeta(D)}$.\n  By $(\\star)$, there is a $(u,w)$-directed walk of Type~1 or Type~2.\n  If there is a $(u,w)$-directed walk of Type~1, then there is nothing to prove.\n  Now suppose that there exists a $(u,w)$-directed walk $Z_u$ of Type~2.\n  Then $Z_u$ contains vertices which induce a directed hole $H$ of length $4$ in $D$ and each vertex on $H$ belongs to one of two distinct partite sets of $D$.\n  Since the vertices on $C$ belong to three distinct partite sets, there must be an arc between a vertex $x$ on $C$ and a vertex $y$ on $H$.\n  Yet, since no directed walk of Type~2 starting at $v$ exists, $(y,x)$ is an arc in $D$.\n  Now, a $(u,y)$-section of $Z_u$, the arc $(y,x)$, $C$ starting at $x$, an $(x,w)$-section of $Z_v$ form a $(u,w)$-directed walk of Type~1.\n  Therefore we have shown that (i) is true.\n  By applying a similar argument, one can show that (ii) is true in the former case.\n\\end{proof}\n\nLet $D$ be a multipartite tournament\nwith a directed cycle, the sink elimination index $\\zeta(D) \\ge 1$, and\nthe sink sequence $\\left(W_0,\\ldots,W_{\\zeta(D)}\\right)$.\nThen we say that a vertex in $\\bigcup_{i=0}^{j-1}W_i$ is of {\\it Type 1} (resp.\\ {\\it Type 2}) if (i) (resp.\\ (ii)) of the above theorem is true. By the same theorem, each vertex in $\\bigcup_{i=0}^{j-1}W_i$ is of Type~1 or Type~2.\n\n\\begin{Thm}\\label{thm:at most 3}\n  If a $k$-partite tournament $D$ has a sink and a directed cycle for an integer $k \\ge 2$, then the competition period of $D$ is at most three.\n  Especially, if $k=2$, then the competition period of $D$ is at most two.\n\\end{Thm}\n\n\n\\begin{proof}\n  Let $D$ be a $k$-partite tournament with a sink and a directed cycle for an integer $k \\ge 2$.\n  Since $D$ has a sink and a directed cycle, $\\zeta(D) \\ge 1$.\n  Let $\\left(W_0,\\ldots ,W_{\\zeta(D) }\\right)$ be the sink sequence of $D$ and $\\mathcal{U}_{\\zeta(D)}=\\bigcup_{i=0}^{\\zeta(D)-1}W_i$.\n  Since $D$ has a directed cycle, $W_{\\zeta(D)} = \\emptyset$ by Proposition~\\ref{prop:acyclic-digraph}.\n  By Proposition~\\ref{Prop:notempty}, $C^m(D)$ is not empty for every positive integer $m$.\n \n \n  If $u \\in \\mathcal{U}_{\\zeta(D)}$, then $u$ has no $m$-step prey for any integer $m \\ge \\zeta(D)$ by Lemma~\\ref{lem:walk-length}.\n  Therefore every vertex in $\\mathcal{U}_{\\zeta(D)}$ is isolated in $C^m(D)$ for any integer $m \\ge \\zeta(D)$.\n   Thus it is sufficient to consider the vertices in  $V(D_{\\zeta(D)})$ when determining the competition period of $D$.\n\n\n   Suppose that there exist vertices $w_1$ and $w_2$ in\n   $\\mathcal{U}_{\\zeta(D)}$ of Type~1 and Type~2, respectively.\n   Fix $w \\in \\mathcal{U}_{\\zeta(D)}$.\n   Then, by Lemma~\\ref{lem:both34}, for each vertex $u$ in $D_{\\zeta(D)}$, there exists a $(u,w)$-directed walk of length $\\ell$ for each integer $\\ell \\ge N_u$ for some positive integer $N_u$.\n   If we denote by $N$ the maximum of such integers $N_u$, then there exists a $(u,w)$-directed walk of length $\\ell$ for each vertex $u$ in $D_{\\zeta(D)}$ and each integer $\\ell \\ge N$.\n   Thus, for each integer $m \\ge N$, $w$ is an $m$-step prey of each vertex $u$ in $D_{\\zeta(D)}$ and so $V(D_{\\zeta(D)})$ forms a clique in $C^m(D)$ and $D$ has the competition period one.\n\n   Now it remains to consider the case that every vertex in $\\mathcal{U}_{\\zeta(D)}$ is only of Type~1 or every vertex in $\\mathcal{U}_{\\zeta(D)}$ is only of Type~2.\n   We first consider the case in which every vertex in $\\mathcal{U}_{\\zeta(D)}$ is only of Type~1.\nSuppose that there are two vertices $u_1$ and $u_2$ in $D_{\\zeta(D)}$ which are adjacent in infinitely many step competition graphs of $D$.\n   Then there exist a $(u_1,w)$-directed walk $Z_1$ and a $(u_2,w)$-directed walk $Z_2$ of the same length $\\ell(u_1,u_2) \\ge |V(D)|$ for a vertex $w$ in $D$.\n   If $w \\in V(D_{\\zeta(D)})$, then $u_1$ and $u_2$ are adjacent in $C^m(D)$ for each integer $m \\ge \\ell(u_1,u_2)$ since $D_{\\zeta(D)}$ has no sink.\n   Now suppose that $w \\in \\mathcal{U}_{\\zeta(D)}$.\n   Since $\\ell(u_1,u_2) \\ge |V(D)|$, each of $Z_1$ and $Z_2$ is of Type~1 or Type~2 by Lemma~\\ref{lem:34cycle1}.\n   By the case assumption, there exist a $(u_1,w)$-directed walk $Z_3$ and a $(u_2,w)$-directed walk $Z_4$ of Type~1.\n  Then, by considering the following three cases:\n  \\begin{itemize}\n    \\item[(a)] both of $Z_1$ and $Z_2$ are of Type~1;\n    \\item[(b)] both of $Z_1$ and $Z_2$ are of Type~2;\n    \\item[(c)] $Z_1$ and $Z_2$ are of different types\n  \\end{itemize}\n   and by applying Lemmas~\\ref{lem:3cycle} and \\ref{lem:both34} to $Z_1, Z_2, Z_3$, or $Z_4$ whichever suitable,\n   we may deduce one of the following:\n  \\begin{itemize}\n   \\item[(i)] for some positive integer $L(u_1,u_2)$ and any integer $m \\ge L(u_1,u_2)$,\n   $u_1$ and $u_2$ have an $(\\ell(u_1,u_2)+3m)$-step common prey;\n   \\item[(ii)]  $u_1$ and $u_2$ have an $m$-step common prey for any integer $m \\ge N(u_1,u_2)$ for some positive integer $N(u_1,u_2)$.\n   \\end{itemize}\nSuppose (i) happens and $u_1$ and $u_2$ have an $\\ell(u_1,u_2)+3m^*+i$ prey for some integer $m^* \\ge L(u_1,u_2)$ and some $i$ in $\\{1,2\\}$.\nThen, by repeating the above argument for $\\ell(u_1,u_2)+3m^*+i$ instead of $\\ell(u_1,u_2)$, we may guarantee the existence of a positive integer $L'(u_1,u_2) \\ge L(u_1,u_2)$ such that $u_1$ and $u_2$ have an $(\\ell(u_1,u_2)+3m^*+i + 3m)$-step common prey for any integer $m \\ge L'(u_1,u_2)$.\nNow $u_1$ and $u_2$ have an $(\\ell(u_1,u_2)+3m)$-step common prey and an $(\\ell(u_1,u_2)+3m +i)$-step common prey for any integer $m \\ge L'(u_1,u_2)$.\nEven if $u_1$ and $u_2$ have an $(\\ell(u_1,u_2)+3m^{**}+j)$-step common prey for some $m^{**} \\ge L'(u_1,u_2)$ and some $j \\in \\{1,2\\} \\setminus i$, we may apply the same argument to find a positive integer $L''(u_1,u_2) \\ge L'(u_1,u_2)$ such that $u_1$ and $u_2$ have an $(\\ell(u_1,u_2)+3m)$-step common prey, an $(\\ell(u_1,u_2)+3m +i)$-step common prey, and an $(\\ell(u_1,u_2)+3m +j)$-step common prey for any integer $m \\ge L''(u_1,u_2)$.\n\nWe let $M(u_1,u_2)$ stands for one of  $L(u_1,u_2)$, $L'(u_1,u_2)$, $L''(u_1,u_2)$, $N(u_1,u_2)$, whichever appropriate.\nLet $L$ be the maximum of $M(u_1,u_2)$ over the pairs $\\{u_1,u_2\\}$ in $D_{\\zeta(D)}$ which are adjacent in infinitely many step competition graphs of $D$ ($L$ exists since there are at most $\\binom{|V({D_{\\zeta(D)}})|}{2}$ pairs to consider).\n   Then it is easy to check that $C^{L+i}(D)=C^{L +3+i}(D)$ for each nonnegative integer $i$, so the competition period of $D$ is $1$ or $3$.\n  Using Lemma~\\ref{lem:4cycle}, one can show that the competition period of $D$ is at most two by a similar argument if every vertex in $\\mathcal{U}_{\\zeta(D)}$ is of Type~2, from which the `especially' part follows.\n \\end{proof}\n\n\n\n\\section{Strongly connected multipartite tournaments}\\label{sec:primitive}\n\nIn this section, we study competition indices of strongly connected multipartite tournaments.\nCho and Kim~\\cite{cho2004competition} showed that a digraph without sinks has competition period $1$.\nIn this section, we show that the competition index of a primitive digraph is at most its exponent.\n\n\\begin{Prop}\\label{prop:adjacent}\n Let $D$ be a digraph without sinks.\n If two vertices are adjacent in $C^M(D)$ for a positive integer $M$, then they are also adjacent in $C^m(D)$ for any positive integer $m \\ge M$.\n\\end{Prop}\n\n\\begin{proof}\n  Let $x$ and $y$ are adjacent in $C^M(D)$.\n  Then $x$ and $y$ have an $M$-step common prey $z$ in $D$.\n  Since $D$ has no sinks, $z$ has an out-neighbor $w$ in $D$.\n  Then $w$ is an $(M+1)$-step common prey of $x$ and $y$.\n  Hence $x$ and $y$ are adjacent in $C^{(M+1)}(D)$.\n  We may repeat this argument to show that $x$ and $y$ are adjacent in $C^{(M+2)}(D)$.\n  In this way, we may show that $x$ and $y$ are adjacent in $C^m(D)$ for any positive integer $m \\ge M$.\n\\end{proof}\n\n\n\n\\begin{Thm}\\label{thm:primitive digraph}\nLet $D$ be a primitive digraph. Then\n\\begin{itemize}\n  \\item[(i)] $C^m(D)$ is a complete graph for each integer $m\\ge\\exp(D)$;\n  \\item[(ii)] ${\\rm cindex}(D)\\le\\exp(D)$;\n  \\item[(iii)] ${\\rm cperiod}(D)=1$.\n\\end{itemize}\n\\end{Thm}\n\\begin{proof}\nSince we assumed that the underlying graph of each digraph $D$ dealt in this paper is simple, $D$ contains neither a loop nor a directed cycle of length $2$.\nThen, by the hypothesis that $D$ is primitive, $|V(D)| \\ge 4$.\nNow take two distinct vertices  $u$ and $v$ in $D$ and let $t=\\exp(D)$.\nThen there exist a $(u,w)$-directed walk and a $(v,w)$-directed walk of length $t$ for any vertex $w$ in $D$.\nTherefore $u$ and $v$ are adjacent in $C^t(D)$.\nSince $u$ and $v$ are arbitrarily chosen, $C^t(D)$ is a complete graph.\nSince $D$ is primitive, $D$ does not contain a sink and so, by Proposition~\\ref{prop:adjacent}, the statement (i) is true.\nBy the definitions of cindex and cperiod, the statements (ii) and (iii) are true.\n\\end{proof}\n\n   Let $D$ be a strongly connected $k$-partite tournament of order $n$ such that the length of a longest directed cycle is $k$ for an integer $4 \\le k \\le n$ and take two vertices $u$ and $v$ in $D$.\n   Bondy~\\cite{bondy1976diconnected} showed that for an integer $k\\ge3$, a strongly connected $k$-partite tournament contains a directed cycle of length $m$ for each integer $3 \\le m \\le k$.\n   Therefore $D$ is primitive.\n   On the other hand, Volkmann~\\cite{volkmann2007multipartite} showed that for an integer $k\\ge3$, every vertex of any strongly connected $k$-partite tournament with a longest cycle of length $k$ belongs to a directed cycle of length $m$ for each integer $3 \\le m \\le k$.\n   Thus there is a directed cycle $C_{m}$ of length $m$ which contains the vertex $u$ for each integer $3 \\le m \\le k$.\n   Now, since $D$ is strongly connected, there is a $(u,v)$-directed walk $W_1$ of length $l \\le n-1$ in $D$.\n  Since $l\\le n-1$, $F(3,4,\\ldots,k) +n -l > F(3,4,\\ldots,k)$.\n  Therefore, by concatenating the directed cycles, we obtain a closed directed walk $W_2$ of length $F(3,4,\\ldots,k) +n -l$.\n  Now $W_2 \\rightarrow W_1$ is a $(u,v)$-directed walk of length $F(3,4,\\ldots,k)+n$.\n  Since $u$ and $v$ were arbitrarily chosen, $\\exp(D) \\le F(3,4,\\ldots,k)+n$.\n  It is known that $F(3,4)=5$ and $F(3,4,\\ldots,k)=2$ for any integer $k \\ge 5$.\n  Thus\n\\begin{equation*}\\exp(D)\\le\n  \\begin{cases}\n    5+n, & \\mbox{if $k=4$;} \\\\\n    2+n, & \\mbox{otherwise.}\n  \\end{cases}\n  \\end{equation*}\n\nNow, by Theorem~\\ref{thm:primitive digraph}(ii), we have the following proposition.\n\\begin{Prop}\n  Let $D$ be a strongly connected $k$-partite tournament of order $n$ such that the length of a longest directed cycle is $k$ for an integer $4 \\le k \\le n$.\n  Then\n  \\begin{equation*}\\mathrm{cindex}(D)\\le\n  \\begin{cases}\n    5+n, & \\mbox{if $k=4$;} \\\\\n    2+n, & \\mbox{otherwise.}\n  \\end{cases}\n  \\end{equation*}\n\\end{Prop}\n\\section{Tournaments}\\label{sec:tournaments}\n\nA tournament with $n$ vertices, denoted by $T_n$, is a digraph resulting from\norienting the edges of a complete graph $K_n$.\nThe outdegree of a vertex $v_i$ in\n$T_n$ is called the score of $v_i$, denoted by $s_i$.\nIf the vertices of an $n$-tournament\n$T_n$ are labeled $v_1,v_2 ,\\ldots, v_n$ so that $0\\le s_1 \\le s_2\\le \\cdots\\le s_n$, then the sequence $(s_1,s_2,\\ldots,s_n)$ is called the {\\em{score sequence}} of $T_n$.\n\nLet $D$ be a $k$-partite tournament with $n$ vertices. Then clearly $k$ is a positive integer with $k\\le n$. One can easily see that $k=n$ if and only if $D$ is a tournament.\n\n\\begin{Thm}\\cite{Ahn1999m}\\label{thm:tournament}\n Let $D$ be a tournament with $n$ vertices and $(s_1,s_2,\\ldots,s_n)$ be its score sequence.\n Then $C^m(D)$ is as follows:\n \\begin{itemize}\n   \\item[(i)] $C^2(D)=\\begin{cases}\n                       K_{n-2} \\cup I_{2}, & \\mbox{if } s_1=0,s_2=1 ;  \\\\\n                       K_{n-1} \\cup I_{1}, & \\mbox{if } s_1=0,s_2\\ge2;\n                       \\\\\n                       \\mbox{$K_{n}$ or $K_{n}-P_2$ or $K_{n}-P_3$}, & \\mbox{if } s_1=1,s_2\\ge2;\n                       \\\\\n                       \\mbox{$K_{n}-P_3$ or $K_{n}-P_4$}, & \\mbox{if } s_1=s_2=1,s_3\\ge2;\n                       \\\\\n                       K_{n}, & \\mbox{if } s_1\\ge2;\n                     \\end{cases}$\n   \\item[(ii)] $C^3(D)=\\begin{cases}\n                       K_{n}, & \\mbox{if either $s_1=1,s_2\\ge2$ or $s_1\\ge2$};  \\\\\n                       K_{n}-P_2, & \\mbox{if } s_1=s_2=1,s_3\\ge2;\n                       \\\\\n                       K_n-C_3, & \\mbox{if } s_1=s_2=s_3=1;\n                     \\end{cases}$;\n   \\item[(iii)] for $m\\ge 4$, $C^m(D)=\\begin{cases}\n                       K_n& \\mbox{if $s_1=1,s_2\\ge2$ or $s_1=s_2=1,s_3\\ge2$ or $s_1\\ge2$};  \\\\\n                       K_n-C_3, & \\mbox{if } s_1=s_2=s_3=1.\n                     \\end{cases}$\n \\end{itemize}\n\\end{Thm}\n\nTheorem~\\ref{thm:tournament} left out the characterization of the $m$-step competition graph of a tournament with sinks for $m \\ge 3$.\nIn this section, we take care of it to eventually compute the competition period and competition index of a tournament with sinks.\n\n\\begin{Prop}\\label{prop:onesink}\n  There is at most one sink in any tournament.\n\\end{Prop}\n\\begin{proof}\n  Suppose to the contrary that there is a tournament $D$ with at least two sinks.\nLet $x$ and $y$ be sinks of $D$.\nSince $D$ is a tournament, one of $(x,y)$ or $(y,x)$ must be in $A(D)$, which is a contradiction.\nHence there is at most one sink in any tournament.\n\\end{proof}\n\n\\begin{Cor}\n Let $n$ be an integer with $n\\ge3$. Then a tournament of order $n$ is acyclic if and only if its sink elimination index is $n-1$.\n\\end{Cor}\n\n\n\n\\begin{Thm}\\label{thm:tournament2}\n  Let $D$ be a tournament of order $n \\ge 2$ with a sink and let $\\zeta(D)$ be the sink elimination index of $D$.\n  Then, for a positive integer $m$, the following are true:\n  \\begin{itemize}\n    \\item[(i)] $1 \\le \\zeta(D) \\le n-1$ with $\\zeta(D) \\neq n-2$.\n        Moreover, for each integer $i$ satisfying $1 \\le i \\le n-1$ and $i \\neq n-2$, there exists a tournament with the sink elimination index $i$;\n    \\item[(ii)] if $1 \\le m < \\zeta(D)$, then $C^m(D)$ is the union of the complete graph with vertex set $V(D_{m-1}) \\setminus W_{m-1}$ and the empty graph with the vertex set $\\bigcup_{i=0}^{m-1}{W_i}$;\n  \\item[(iii)] if $m \\ge \\zeta(D)$, then $C^m(D)$ is the union of the complete graph with vertex set $V(D_{\\zeta(D)})$ and the empty graph with the vertex set $\\bigcup_{i=0}^{\\zeta(D)-1}{W_i}$;\n \n    \\item[(iv)] $\\mathrm{cperiod}(D)=1$;\n    \\item[(v)] $\\mathrm{cindex}(D)=\\zeta(D)$.\n  \\end{itemize}\n\\end{Thm}\n\\begin{proof}\n  Let $(W_0,\\ldots,W_{\\zeta(D)})$ be the sink sequence of $D$.\n  By the hypothesis that $D$ has a sink, $W_0 \\neq \\emptyset$.\n  Then $|W_0| =1$ by Proposition~\\ref{prop:onesink}.\n  Since $n \\ge 2$, $W_0 \\neq V(D)$.\n  Thus $\\zeta(D) \\neq 0$ and so $1 \\le \\zeta(D) \\le n-1$.\n  Now $|W_i| =1$ for each integer $0 \\le i \\le \\zeta(D)-1$ by Proposition~\\ref{prop:onesink}.\n\n  Suppose $\\zeta(D)=n-2$.\n  Then $D_{\\zeta(D)}$ is a tournament of order $2$, which has a sink and a non-sink, and we reach a contradiction.\n  Therefore $\\zeta(D) \\neq n-2$.\n\n   To show the `moreover' part of the statement (i), fix $i$ such that $1 \\le i \\le n-1$ and $i \\neq n-2$.\n   We consider a tournament $D'$ defined by\n $V(D')=\\{v_1,v_2,\\ldots,v_n\\}$ and $A(D')=\\{(v_l,v_k) \\mid 1 \\le k < l \\le n\\} \\setminus \\{(v_n,v_{i+1})\\} \\cup \\{(v_{i+1},v_{n})\\}$.\n It is easy to check that $W_{j-1}=\\{v_{j}\\}$ for each $j = 1, \\ldots,  i$ and so $\\zeta(D') \\ge i$.\n On the other hand, there exists a directed cycle $v_{n}\\to  v_{n-1} \\to v_{n-2} \\cdots \\to v_{i+1} \\to v_n$ in $D'$ and so we may conclude that $\\zeta(D') = i$.\nThus the statement (i) is true.\n\nTo show the statements (ii) and (iii), we denote the vertices of $D$ as  $v_1,v_2,\\ldots,v_n$ so that $W_{i} = \\{v_{i+1}\\}$ for each integer $0 \\le i \\le \\zeta(D)-1$.\n Then, for each integer $1 \\le i \\le \\zeta(D)$, the length of a longest directed walk with an initial vertex $v_i$ is $i-1$ by Lemma~\\ref{lem:walk-length}.\nTake a positive integer $m$.\n Suppose $1 \\le m < \\zeta(D)$.\n  Then, for every vertex $v$ in $V(D_{m-1}) \\setminus W_{m-1}$, there is an arc from $v$ to $v_m$.\n  Concatenating this arc with the directed walk of length $m-1$ from $v_m$ to $v_1$, every vertex in $V(D_{m-1}) \\setminus W_{m-1}$ has $v_1$ as an $m$-step prey.\n   Since $V(D_{m-1}) \\setminus W_{m-1} = V(D) \\setminus \\bigcup_{i=0}^{m-1}{W_i}$, $V(D_{m-1}) \\setminus W_{m-1}$ forms a clique of size $|V(D) \\setminus \\bigcup_{i=0}^{m-1}{W_i}| = n-m$ in $C^m(D)$.\n   Since no vertex in $\\bigcup_{i=0}^{m-1}{W_i}$ has an $m$-step prey in $D$ by Lemma~\\ref{lem:walk-length}, the vertices in $\\bigcup_{i=0}^{m-1}{W_i}$ are isolated in $C^m(D)$.\n   Thus the statement (ii) is true.\n\n   Suppose $m \\ge \\zeta(D)$.\n   By (i), $1 \\le \\zeta(D) \\le n-1$ and $\\zeta(D) \\neq n-2$.\n If $\\zeta(D)=n-1$, then there is at most one vertex which has an $m$-step prey in $D$ and so $C^m(D)=I_{n}=K_{1} \\cup I_{n-1}$.\n Suppose $1 \\le \\zeta(D) \\le n-3$.\n    Then $D_{\\zeta(D)}$ is a tournament of order at least $3$ and so $W_{\\zeta(D)} \\neq V(D_{\\zeta(D)})$ by Proposition~\\ref{prop:onesink}.\n  By the definition of $\\zeta(D)$, $W_{\\zeta(D)}=\\emptyset$, that is, $D_{\\zeta(D)}$ is a tournament without sinks.\n  Thus, for each vertex $v \\in D_{\\zeta(D)}$, there exists a directed walk $X_v$ in $D_{\\zeta(D)}$ of length $k$ from $v$ to a vertex in $D_{\\zeta(D)}$ where $k=m-\\zeta(D)$.\n  Since $D_{\\zeta(D)}$ is a subdigraph of $D$, $X_v$ is a directed walk of length $k$ in $D$.\n  By the definition of sink sequence, there is a directed walk of length $\\zeta(D)$ from each vertex in  $D_{\\zeta(D)}$ to $v_1$ in $D$.\n  By concatenating those two directed walks, we obtain a $(v,v_1)$-directed walk of length $m$ for each vertex $v \\in D_{\\zeta(D)}$ in $D$.\n  Hence the vertices in $D_{\\zeta(D)}$ forms a clique of size $n-\\zeta(D)$ in $C^m(D)$.\n  Since no vertex in $\\bigcup_{i=0}^{\\zeta(D)-1}{W_i}$ has an $m$-step prey in $D$ by Lemma~\\ref{lem:walk-length}, the vertices in $\\bigcup_{i=0}^{\\zeta(D)-1}{W_i}$ are isolated in $C^m(D)$.\n  Therefore the statement (iii) is valid.\n  Thus the statements (iv) and (v) immediately follow from (ii) and (iii).\n\\end{proof}\n\n\\section{Acknowledgement}\nThis research was supported by the National Research Foundation of Korea(NRF)  (NRF-2017R1E1A1A03070489 and 2016R1A5A1008055) funded by the Korea government(MSIP).\n\n\n\n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\n\\section{Introduction}\n\n\nExperimental observations and theoretical studies of heavy quarkonium $Q \\bar\nQ$ states have played a very valuable role in elucidating the properties of\nquantum chromodynamics (QCD).  A heavy quark is one whose mass, $m_Q$, is\nlarge compared with $\\Lambda_{QCD} \\sim 0.3$, so that the running QCD coupling\n$g_s(M_Q)$ and the associated quantity $\\alpha_s(m_Q) = g_s(M_Q)^2\/(4\\pi)$ are\nreasonably small, allowing perturbative treatments of at least some parts of\nthe physics of $Q \\bar Q$ states and decays.  Furthermore, for\n$m_Q \\gg \\Lambda_{QCD}$, one can obtain an approximate description of many\nproperties of the $Q \\bar Q$ states using nonrelativistic methods, including\npotential models.  From the time of the discovery of\nthe $J\/\\psi$ at BNL \\cite{Aubert:1974js} and SLAC\/SPEAR \\cite{psi_slac} in\n1974, and the $\\Upsilon$ at Fermilab in 1977 \\cite{Herb:1977ek,Innes:1977ae},\nthere has been a steadily growing wealth of data on the various $Q\\bar Q$\nstates, where the $Q$ denotes a charm quark $c$ or a bottom\/beauty quark $b$,\nas well as data on mesons and baryons with charm and\nbottom\/beauty quantum numbers.  Some reviews of heavy quarkonia and\nreferences to the literature include \\cite{quiggrosner79}-\\cite{rosner2013}.\n\nThese experimental achievements motivate the continued theoretical study of the\nstructure and properties of $c \\bar c$ and $b \\bar b$ quarkonium states.  Among\nquarkonium decays, radiative decays are particularly valuable as tests of\nvarious models, since the photon is directly observed and the nature of the\nelectromagnetic transition is well understood. One of the simplest types of\nradiative decays is the electric dipole (E1) transition between a $Q \\bar Q$\nstate with radial quantum number $n$ and spectroscopic type $n \\, {}^{2S+1} \\,\nL_J =n {}^3 \\, L_J$ with $L=1$ (P-wave) and $J=0, \\ 1, \\ 2$, denoted\n$\\chi_{QJ}(nP)$ in standard notation, where $Q=c, \\ b$, and a lower-lying $Q\n\\bar Q$ S-wave state $n' \\, {}^3S_1$, in particular, $J\/\\psi$ and\n$\\psi(2S)$ for the $c \\bar c$ system, and $\\Upsilon(n'S)$ with $n'=1, \\ 2, \\ 3$\nfor the $b \\bar b$ system. In terms of $J^{PC}$ values, these decays are of the\nform $J^{++} \\to 1^{--} + \\gamma$, where $J=0, \\ 1, \\ 2$.  The P-wave $c\\bar c$\nstates were first observed in 1976 by the SLAC-LBL experiment at SLAC\/SPEAR\n\\cite{Whitaker:1976hb,Biddick:1977sv}.  The P-wave $b \\bar b$ states were first\nobserved by the Columbia-Stony Brook (CUSB) experiment at the Cornell CESR\n$e^+e^-$ storage ring \\cite{klopfenstein83,pauss83} and confirmed by the CLEO\nexperiment at CESR \\cite{haas84}.  Larger data samples and quite accurate\nmeasurements of branching ratios for radiative decays of P-wave $b \\bar b$\nstates were obtained later, in particular, by the CLEO III \\cite{kornicer2011}\nand BABAR experiments \\cite{babar2014}.  Valuable results have also been\nobtained from hadron colliders, including the observation of the\n$\\chi_{bJ}(3P)$ $b \\bar b$ states at the Large Hadron Collider (LHC)\n\\cite{Aad:2011ih} and the measurement of the mass of $\\chi_{b1}(3P)$ by LHCb\n\\cite{Aaij:2014hla}.\n\nThere have been a number of theoretical studies of these\nE1 transitions based on a range of different models\n\\cite{Karl:1980wm}-\\cite{Brambilla:2005zw}.  Many of these models make use of\nnonrelativistic potentials, such as the potential $V = -(4\/3)\\alpha_s(m_Q)\/r +\n\\sigma r$, where the first term is a non-Abelian Coulomb potential representing\none-gluon exchange at short distances and the second term is the linear\nconfining potential, with $\\sigma$ denoting the string tension.  These are\nreasonable models, since a $Q \\bar Q$ bound-state system is nonrelativistic if\n$m_Q\/\\Lambda_{QCD} \\gg 1$.\n\nIt is of interest to investigate these radiative decays of P-wave quarkonium\nstates using a different type of model, namely the light-front quark model\n(LFQM)\\cite{Terentev:1976jk}-\\cite{Cheng:2003sm}.  This approach permits a\nfully relativistic treatment of the quark spins and the internal motion of the\nconstituent quarks.  In this covariant approach, the hadronic structure for\nsmall momentum transfer is represented by one-loop diagrams evaluated on the\nlight cone. It has been used to study semileptonic and nonleptonic decays of\nheavy-flavor $D$ and $B$ mesons and also to evaluate radiative decay rates of\nheavy mesons \\cite{Hwang:2006cua,Choi:2007se,Hwang:2010iq,Ke:2010vn,Ke:2013zs}.\nIn particular, in \\cite{Ke:2013zs} with Ke and Li, we used this approach to\ncalculate the widths for the radiative decays of heavy $0^{++}$ and $1^{+-}$ $Q\n\\bar Q$ mesons.\n\nIn the light-front formalism, one chooses the coordinate where $q^{+}=0$, in\nwhich the quark current cannot create or annihilate pairs, and the relevant\ntransition matrix element can be computed as an overlap of Fock-space\nwavefunctions. The terms involving pair production or annihilation vanish\n\\cite{Drell:1969km,Brodsky:1997de}.  An advantage of the\nlight-front quark model is that it is manifestly covariant. In the light-front\napproach, it is easy to boost a hadron bound states from one inertial Lorentz\nframe to another one when the bound state wavefunction is known in a particular\nframe\\cite{Brodsky:1997de}.\n\n\nIn this paper, extending our previous work with Ke and Li in Ref.\n\\cite{Ke:2013zs}, we study the radiative decays\n\\beq\n\\chi_{c1}(1P) \\to J\/\\psi + \\gamma\n\\label{chic1}\n\\eeq\nand\n\\beq\n\\chi_{b1}(nP) \\to \\Upsilon(n'S) + \\gamma\n\\label{chibj}\n\\eeq\nwhere $n \\ge n'$ by using the light-front quark model. With the front-front\nformalism, we perform a numerical calculation the widths for these decays and\nthen compare the results with theoretical calculations based on other\napproaches.\n\nThe paper is organized as follows: In Section \\ref{LFQM}, we derive the\nformulas for the radiative decay $1^{++} \\to 1^{--} +\\gamma$. Then in section\n\\ref{WF}, we discuss the meson wavefunctions that are relevant to the\nlight-front approach. In Section \\ref{ANALYSIS}, we discuss numerical results\nfor the decay widths of $\\chi_{c1}(1P) \\to J\/\\psi + \\gamma$ and\n$\\chi_{b1}(nP) \\to \\Upsilon(n'S) + \\gamma$. Our conclusions are given in\nSection \\ref{CON}.\n\n\n\n\\section{Light-front formalism for the decays $1^{++}  \\to 1^{--} +\\gamma$}\n\\label{LFQM}\n\n\n\\subsection{Notation}\n\nHere we briefly summarize the notation that is relevant for radiative\ntransition of meson. We follow the covariant light-front approach of\n\\cite{Jaus:1999zv,Cheng:2003sm} and use the same notation. In light-front\ncoordinates, a (four-)momentum $p$ is expressed as\n\\beq\np^\\mu = (p^{-},p^{+},{\\vec p}_{\\perp})\n\\label{p}\n\\eeq\nwhere\n\\beq\np^{+}=p^0+p^3, \\quad p^{-}=p^0-p^3\n\\label{ppm}\n\\eeq\nand\n\\beq\n{\\vec p}_{\\perp}=(p^1,p^2) \\ .\n\\label{pperp}\n\\eeq\nThus,\n\\beqs\np^2 & = & (p^0)^2 - |\\vec p|^2 = (p^0)^2 - (p^3)^2 - |{\\vec p}_\\perp|^2 \\cr\\cr\n    & = & p^+ p^- -|{\\vec p}_\\perp|^2 \\ .\n\\label{psq}\n\\eeqs\nWe denote the momentum of the parent (incoming) meson as $P'=p'_1 +p_2$, where\n$p'_{1}$ and $p_2$ are the momenta of the constituent quark and anti-quark,\nwith mass $m'_1$ and $m_2$, respectively.  Similarly, we label the momentum of\nthe daughter (outgoing) quarkonium meson as $P''=p''_1+p_2$, where\n$p''_{1}$ is the momentum of the constituent quark, with mass $m''_1$.  For our\napplication to $Q \\bar Q$ quarkonium systems, $m'_1=m_2 = m''=m_Q$.  The\nfour-momentum of the parent meson with mass $ M'$, in terms of light-front\ncoordinates, is\n\\beq\nP'=(P'^{-},P'^{+},{\\vec P}'_{\\perp})\n\\label{pp}\n\\eeq\nso $P'^2=P'^{+}P'^{-}-|{\\vec P}_{\\perp}|^2=M'^2$. Similarly, for the outgoing\nmeson, $P''^2=M''^2$.  In what follows, the vector signs on transverse momentum\ncomponents are to be understood implicitly and are suppressed in the notation.\nThe internal motion of the constituents can be described\nby the variables $(x_2,p'_{\\perp})$, where\n\\beqs\n&&p'^{+}_1=x_1P'^{+}, \\quad  p^{+}_2=x_2P'^{+} \\nonumber\\\\\n&&p'_{1\\perp}=x_1P'_{\\perp}+p'_{\\perp}, \\quad\n  p_{2\\perp}=x_2P'_{\\perp} - p'_{\\perp} \\nonumber\\\\\n&&x_1 + x_2 =1\n\\label{parts}\n\\eeqs\nand $p''_{\\perp}$ can be expressed as\n\\beq\np''_{\\perp}=p'_{\\perp}-x_2 q_{\\perp} .\n\\label{ppp}\n\\eeq\n\n\n\\subsection{Form factors}\n\nLet us define $P=P'+P ''$ and $q=P'-P''$.  Since the\ninitial P-wave $1^{++}$ $Q\\bar Q$ state is an axial-vector, we denote it as\n$A$, while the final $1^{--}$ $Q \\bar Q$ state is a vector, denoted $V$.\nThe general amplitude for the transition $A (1^{++}) \\to V(1^{--}) +\\gamma$\nhas the form\n\\beq\ni{\\cal A} \\left(A (P') \\to  V(P'') \\gamma (q) \\right)\n  =\n\\varepsilon^{*}_{\\mu}(q) \\varepsilon'_{\\nu}(P')\\varepsilon''^{*}_{\\rho}(P'')i\n\\tilde {\\cal A }^{\\mu\\nu\\rho} \\ ,\n\\label{amp}\n\\eeq\nwhere $\\varepsilon'_{\\nu}(P')$, $\\varepsilon''^{*}_{\\rho}(P'')$, and\n$\\varepsilon^{*}_{\\mu}(q)$ are the polarization (four-)vectors of the parent\nheavy axial-vector meson, the daughter heavy vector meson, and the photon,\nrespectively.  The structure of this amplitude was given in\n\\cite{Dudek:2006ej}.  We review this next. Since quantum\nelectrodynamic (QED) interactions are invariant under parity and time reversal\n(and thus also CP), this amplitude must be P- and T-invariant. In addition to\nthese two conditions, the transverse properties of the polarization vectors\nyield the three further conditions\n\\beq\n\\varepsilon'_{\\nu}(P') (P+q)^{\\nu}=0,\n\\label{edota}\n\\eeq\n\\beq\n\\varepsilon''^{*}_{\\rho}(P'') (P-q)^{\\rho}=0,\n\\label{edobv}\n\\eeq\nand\n\\beq\n\\varepsilon^{*}_{\\mu}(q)q^{\\mu}=0.\n\\label{ephoton}\n\\eeq\nCondition (\\ref{ephoton}) is also implied by electromagnetic gauge invariance.\nApplying these conditions, we obtain the following general amplitude (to be\nsimplified below):\n\\beqs\ni\\tilde {\\cal A }^{\\mu\\nu\\rho} &=&\nf_1 \\epsilon^{\\mu\\nu\\rho\\alpha}P_{\\alpha} +\nf_2 \\epsilon^{\\mu\\nu\\rho\\alpha}q_{\\alpha} +\nf_3 \\epsilon^{\\rho\\nu \\alpha \\beta}P^\\mu P_\\alpha q_\\beta\\nonumber\\\\\n&+&\nf_4 \\epsilon^{\\mu\\nu \\alpha \\beta}P^\\rho P_\\alpha q_\\beta +\nf_5 \\epsilon^{\\rho\\mu \\alpha \\beta}P^\\nu P_\\alpha q_\\beta  \\ .\n\\eeqs\nThis expression can be simplified by using the fact that the photon only has\ntwo transverse polarization states, so the timelike component\n$\\varepsilon^{*}_{0}(q)=0$. Taking the parent axial vector meson $A(P')$ to be\nin its rest frame, we have $P'^{\\mu}=(M',0)$, where $M'$ is mass of $A(P')$.\nThe $f_3$ term can be eliminated:\n\\beqs\n&&[f_3 \\epsilon^{\\rho\\nu \\alpha \\beta}P^\\mu P_\\alpha q_\\beta]\n\\varepsilon^{*}_{\\mu}(q) \\varepsilon'_{\\nu}(P')\\varepsilon''^{*}_{\\rho}(P'')\n \\propto  \\varepsilon^{*}_{\\mu}(q)P^{\\mu} \\nonumber\\\\\n&&= 2\\varepsilon^{*}_{\\mu}(q)P'^{\\mu}=2\\varepsilon^{*}_{0}(q)P'^{0}=0 \\ .\n\\label{f3z}\n\\eeqs\nFurthermore, the $f_1$ term vanishes due to electromagnetic gauge invariance,\n$q_{\\mu}\\tilde A^{\\mu\\nu\\rho}=0$. Therefore, the general\namplitude that satisfies the five conditions above is given by\n\\cite{Dudek:2006ej}:\n\\beqs\n&&i\\tilde {\\cal A }^{\\mu\\nu\\rho} \\to i { A }^{\\mu\\nu\\rho} \\nonumber\\\\\n&& =\nf_2 \\epsilon^{\\mu\\nu\\rho\\alpha}q_{\\alpha} +\nf_4 \\epsilon^{\\mu\\nu \\alpha \\beta}P^\\rho P_\\alpha q_\\beta +\nf_5 \\epsilon^{\\rho\\mu \\alpha \\beta}P^\\nu P_\\alpha q_\\beta  \\ . \\nonumber \\\\\n\\label{formfactor}\n\\eeqs\nThe $f_2$ term corresponds to the electric dipole (E1) transition and makes the\ndominant contribution to the amplitude, while the $f_4$ and $f_5$ terms\ncorrespond to the magnetic dipole (M2) transition and make subdominant\ncontributions \\cite{Cho:1994gb,Shao:2012fs}. A detailed analysis of parity and\ntime-reversal invariance of this general amplitude is given in\nAppendix \\ref{PT invariance}.\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{0.45\\textwidth}{!}\n\\includegraphics{PICTRAD.eps}\n\n\\caption {Feynman diagrams for radiative transitions in the light-front\nframework.}\n\\label{p1}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Calculation of radiative decay amplitude}\n\nIn general, the width for an electromagnetic dipole transition between an\ninitial $Q \\bar Q$\nstate $n \\, {}^3P_J$ and a final state $n' \\, {}^3S_1 + \\gamma$\nis given by (e.g., \\cite{kwong_rosner_quigg1987})\n\\beq\n\\Gamma(n \\, {}^3P_J \\to n' \\, {}^3S_1 + \\gamma) =\n\\frac{4\\alpha_{em} e_Q^2 E_\\gamma^3}{9} \\, |\\langle f | {\\vec r} | i \\rangle\n|^2 \\ ,\n\\label{gammageneral}\n\\eeq\nwhere $e_Q$ is the quark of the quark $Q$, $E_\\gamma$ is the energy of the\noutgoing photon in the parent rest frame, and $i$ and $j$ denote the initial-\nand final-state wavefunctions.  For our calculation in the LFQM, we note that\nthe vertex function for the parent axial-vector meson $A(1^{++})$ is given\nby \\cite{Cheng:2003sm}:\n\\beq\niH'_{A} \\left [ \\gamma^{\\nu} + \\frac{(p'_1-p_2)^{\\nu}}{W'_{A}}  \\right] \\gamma^5 ,\n\\eeq\nand the vertex function for the daughter vector meson $V(1^{--})$ is\n\\beq\niH''_{V} \\left [ \\gamma^{\\rho} - \\frac{(p''_1-p_2)^{\\rho}}{W''_{V}} \\right] \\ ,\n\\eeq\nwhere $H'_{A}$ and $H''_{V}$ are functions of $p'_1$, $p''_1$ and $p_2$.\nThe explicit forms for these vertex functions will be discussed below.\n\nThere are two diagrams that contribute at leading order to the\n$A \\to V + \\gamma$ transition amplitude, so we write\n\\beq\ni{\\cal A }^{\\mu\\nu\\rho}(A \\to V + \\gamma)=i{\\cal A }^{\\mu\\nu\\rho}(a)+i{\\cal A }^{\\mu\\nu\\rho}(b) \\ ,\n\\eeq\nwhere the left-hand diagram in Fig.(\\ref{p1}) corresponds to\n${\\cal A }^{\\mu\\nu\\rho}(a)$ and the right-hand diagram in Fig.(\\ref{p1})\ncorresponds to ${\\cal A }^{\\mu\\nu\\rho}(b)$.  These are related by charge\nconjugation. For the left-hand diagram, the transition amplitude is given by\n\\beq\ni{\\cal A }^{\\mu\\nu\\rho}(a)= e N_{e1'}  \\frac{ N_c}{(2\\pi)^4} \\int d^4p'_1 \\frac{H'_{A} H''_{V}}{N'_1N''_1 N_2} {\\cal S}^{\\mu\\nu \\rho}_a \\ ,\n\\eeq\nwhere\n\\beqs\n{\\cal S}^{\\mu\\nu \\rho}_a  &=& \\text {Tr} \\left\\{ (\\pslash''_1 +m''_1)\\gamma^{\\mu} (\\pslash_1' +m'_1)[\\gamma^{\\nu}+\\frac{(p'_1-p_2)^{\\nu}}{W'_{A}}]\\right. \\nonumber \\\\\n&\\times& \\left. \\gamma^{5} (-\\pslash_2 + m_2) [\\gamma^{\\rho}-\\frac{1}{W''_{V}}(p''_1-p_2)^{\\rho}] \\right\\} \\nonumber \\\\\n&=& -p''^{\\mu}_1 \\text{Tr} [\\pslash'_1 \\gamma^{\\nu}\\pslash_2\\gamma^{\\rho}\\gamma^5  ] + (p'_1 \\cdot p''_1 - m'_1 m''_1)\\nonumber\\\\\n&\\times & \\text{Tr} [ \\gamma^{\\mu}\\gamma^{\\nu}\\pslash_2 \\gamma^{\\rho}\\gamma^5  ] - p'^{\\mu}_1 \\text{Tr}[\\pslash''_1\\gamma^{\\nu}\\pslash_2\\gamma^{\\rho} \\gamma^5] \\nonumber \\\\\n&-& \\frac{m''_1}{W'_{A}}(p'_1-p_2)^{\\nu} \\text {Tr} [\\gamma^{\\mu}\\pslash'_1\\pslash_2\\gamma^{\\rho}\\gamma^5] - m''_1m_2\\nonumber\\\\\n&\\times& \\text {Tr} [\\gamma^{\\mu}\\pslash'_1\\gamma^{\\nu}\\gamma^{\\rho}\\gamma^5] -      \\frac{m'_1}{W'_{A}}(p'_1-p_2)^{\\nu} \\text {Tr} [\\pslash''_1\\gamma^{\\mu}\\pslash_2\\gamma^{\\rho}\\gamma^5] \\nonumber\\\\\n&-&m'_1m_2 \\text {Tr}[\\pslash''_1\\gamma^{\\mu}\\gamma^{\\nu}\\gamma^{\\rho}\\gamma^5]-\\frac{m'_1}{W''_{V}}(p''_1-p_2)^{\\rho} \\nonumber\\\\\n&\\times&  \\text {Tr}[\\pslash''_1\\gamma^{\\mu}\\gamma^{\\nu}\\pslash_2\\gamma^5]- \\frac{m''_1}{W_{V}''}(p''_1-p_2)^{\\rho} \\text {Tr}[\\gamma^{\\mu}\\pslash'_1\\gamma^{\\nu}\\pslash_2\\gamma^5] \\nonumber\\\\\n&-&[\\frac{(p'_1-p_2)^{\\nu}}{W'_{A}}m_2+p_2^{\\nu}] \\text {Tr}[\\pslash''_1\\gamma^{\\mu}\\pslash'_1 \\gamma^{\\rho}\\gamma^5] \\nonumber\\\\\n&-&[\\frac{(p'_1-p_2)^{\\nu}}{W'_{A}}\\frac{(p''_1-p_2)^{\\rho}}{W''_{V}}-g^{\\rho\\nu}]  \\text {Tr}[\\pslash''_1\\gamma^{\\mu}\\pslash'_1\\pslash_2\\gamma^5]  \\nonumber\\\\\n&-&[p^{\\rho}_2+\\frac{m_2}{W''_{V}}(p''_1-p_2)^{\\rho}]\\text\n{Tr}[\\pslash''_1\\gamma^{\\mu}\\pslash'_1\\gamma^{\\nu}\\gamma^5]  \\ ,\n\\label{sintegral}\n\\eeqs\n\\beq\nN'_1=p'^2_1-m'^2_1+i\\epsilon,\n\\label{nprime}\n\\eeq\n\\beq\nN''_1=p''^2_1-m''^2_1+i\\epsilon,\n\\label{ndoubleprime}\n\\eeq\n\\beq\nN_2=p^2_2-m^2_2+i\\epsilon,\n\\label{n2}\n\\eeq\nand $N_{e1'(e2)}$ denotes the electric charge of the constituent quark in units\nof $e$.  The contribution to the amplitude from the right-hand diagram follows\nfrom this.\n\nTo calculate the amplitude in the covariant light-front approach, we need to\nintegrate over the internal momentum, $p'^{-}_1$. In order to do this, we first\nexpress the amplitude in terms of internal momentum $p'_1$ and external\nmomenta $P$ and $q$, as well as $N'_1$, $N''_1$, $N_2$, by using the\nfollowing relations:\n\\beqs\np''_1 &=& p'_1-q \\nonumber\\\\\np_2 &=& (P+q)\/2-p'_1 \\nonumber\\\\\n2p'_1\\cdot p_2 &=& M'^2-N'_1-m'^2_1-N_2-m^2_2  \\nonumber\\\\\n2p''_1\\cdot p_2 &=& M''^2-N''_1-m''^2_1-N_2-m^2_2   \\nonumber\\\\\n2p'_1\\cdot p''_1 &=& -q^2 + N'_1+m'^2_1+N''_1+m''^2_1.\n\\label{relations}\n\\eeqs\nAfter the integration over $p'^{-}_{1}$, one makes the following replacement\n\\cite{Jaus:1999zv,Cheng:2003sm}:\n\\beqs\n\\int d^4p'_1 \\frac{H'_{A} H''_{V}}{N'_1N''_1 N_2} {\\cal S}^{\\mu\\nu \\rho}_a \\varepsilon^{*}_{\\mu} \\varepsilon'_{\\nu}\\varepsilon''^{*}_{\\rho}\\to \\nonumber \\\\\n-i \\pi \\int dx_2 d^2p'_{\\perp} \\frac{h'_{A}h''_{V}}{x_2 \\hat N'_1\\hat\n  N''_1}\\hat{ \\cal S}^{\\mu\\nu \\rho}_a \\hat \\varepsilon^{*}_{\\mu} \\hat\n\\varepsilon'_{\\nu} \\hat \\varepsilon''^{*}_{\\rho} \\ ,\n\\label{replacement}\n\\eeqs\nwhere\n\\beqs\n&&N'_1 \\to \\hat N'_1= x_1 (M'^2-M'^2_{0}) \\nonumber \\\\\n&&N''_1 \\to \\hat N''_1= x_1 (M''^2-M''^2_{0})  \\nonumber \\\\\n&& H'_{A} \\to h'_{A}=(M'^2-M'^2_{0})  \\sqrt{\\frac{x_1 x_2}{N_c}} \\frac{\\tilde M'_{0}}{4M'_{0}} \\phi_{np} (x_2,p'_{\\perp})\\nonumber\\\\\n&& H''_{V} \\to h''_{V}=(M''^2-M''^2_{0})  \\sqrt{\\frac{x_1 x_2}{N_c}} \\frac{1}{\\sqrt{2}\\tilde M''_{0}} \\phi_{n's} (x_2,p''_{\\perp}) \\nonumber\\\\\n&&W'_{A} \\to w'_{A} =  \\frac{\\tilde M'^2_0}{m'_1 - m_2} \\nonumber\\\\\n&& W''_{V} \\to w''_{V} = M''_{0} + m''_1 + m_2 \\ .\n\\label{replacement_relations}\n\\eeqs\nIn the above\nexpressions, $\\phi_{n's}$ and $\\phi_{np}$ represent the wavefunction for the\nS-wave $Q\\bar Q$ meson $V$ and the P-wave $Q\\bar Q$ meson $A$,\nrespectively. We will discuss these wavefunctions in detail in the\nnext section. The definitions of $M'_0$, $M''_0$, $\\tilde M'_0$ and $\\tilde\nM''_0$ are given in appendix \\ref{experssion}. The definition of $\\hat\n\\varepsilon^{*}$, $\\hat \\varepsilon'$ and $\\hat \\varepsilon''^{*}_{\\rho}$ is\ngiven in \\cite{Jaus:1999zv,Cheng:2003sm}.\n\nOne should also include the contribution from zero modes in the $A$ meson.\nIn practice, this amounts to the following replacement for\n$p'_{1\\mu}$ in $\\hat{ \\cal  S}^{\\mu\\nu \\rho}_a $ in the integral\n\\cite{Jaus:1999zv,Cheng:2003sm}:\n\\beqs\n\\hat p'_{1\\mu} &\\to& P_{\\mu} A^{(1)}_1 + q_\\mu A^{(1)}_2 \\ , \\nonumber\\\\\n \\hat p'_{1\\mu} \\hat p'_{1\\nu}  &\\to &g_{\\mu\\nu} A^{(2)}_1 + P_{\\mu}P_{\\nu} A^{(2)}_2  \\nonumber\\\\\n&+& (P_{\\mu}q_{\\nu} + q_{\\mu}P_{\\nu})A^{(2)}_3 + q_{\\mu} q_{\\nu} A^{(2)}_4 \\ , \\nonumber \\\\\n \\hat p'_{1\\mu} \\hat p'_{1\\nu}  \\hat p'_{1\\alpha}  &\\to& (g_{\\mu\\nu}P_{\\alpha} + g_{\\mu\\alpha} P_{\\nu} + g_{\\nu\\alpha}P_{\\mu})A^{(3)}_1 \\nonumber\\\\\n &+& (g_{\\mu\\nu}q_{\\alpha} + g_{\\mu\\alpha} q_{\\nu} + g_{\\nu\\alpha}q_{\\mu})A^{(3)}_2 \\nonumber\\\\\n &+& P_{\\mu}P_{\\nu} P_{\\alpha} A^{(3)}_3 + (P_{\\mu}P_{\\nu}q_{\\alpha} + P_{\\mu}q_{\\nu}P_{\\alpha} \\nonumber\\\\\n &+& q_{\\mu} P_{\\nu} P_{\\alpha}) A^{(3)}_4 + (q_{\\mu}q_{\\nu}P_{\\alpha} + q_{\\mu}P_{\\nu}q_{\\alpha}  \\nonumber\\\\\n &+& P_{\\mu}q_{\\nu} q_{\\alpha}) A^{(3)}_5 + q_{\\mu} q_{\\nu} q_{\\alpha} A^{(3)}_6 \\ .\n\\eeqs\nAfter these operations, the amplitude ${\\cal A }^{\\mu\\nu\\rho}(a)$ can be\nexpressed as a function of the external four-momenta $P$ and $q$. It can be\nparametrized in the following form:\n\\beqs\ni{\\cal A }^{\\mu\\nu\\rho}(a) &=&   f^a_2 \\epsilon^{\\mu\\nu\\rho\\alpha}q_{\\alpha} +  f^a_4 \\epsilon^{\\mu\\nu \\alpha \\beta}P^\\rho P_\\alpha q_\\beta \\nonumber\\\\\n&+& f^a_5 \\epsilon^{\\rho\\mu \\alpha \\beta}P^\\nu P_\\alpha q_\\beta \\  ,\n\\eeqs\n\n\nwith\n\\begin{widetext}\n\\beqs\nf^a_2  (q^2) &=&  e N_{e'_1} \\frac{N_c}{16\\pi^3} \\int dx_2 d^2p'_{\\perp} \\frac{h'_{A}h''_{V}}{x_2\\hat N'_1 \\hat N''_1} (-4)\\cdot\\left [  \\frac{1}{w'_{A}}(m''_1+ m'_1-2m_2) A^{(2)}_1 + \\frac{1}{w''_{V}}(2m_2+m'_1+ m''_1) A^{(2)}_1   \\right. \\nonumber\\\\\n&-& \\left. \\frac{1}{4}(1-2A^{(1)}_2)\\left (-q^2+\\hat N'_1  + \\hat N''_1 + (m'_1 - m''_1)^2\\right) - A^{(1)}_{2} (m''_1 m_2 - m'_1 m_2) -  m'_1 m_2 \\right ] \\ , \\nonumber\\\\\nf^a_4  (q^2)&=&   e N_{e'_1} \\frac{N_c}{16\\pi^3} \\int dx_2 d^2p'_{\\perp} \\frac{h'_{A}h''_{V}}{x_2\\hat N'_1 \\hat N''_1} (-4)\\cdot\\left [\n\\frac{1}{w''_{V}}\\left ( (m'_1 -m''_1) (A^{(2)}_3 + A^{(2)}_4 - A^{(1)}_2) + (m'_1 +m''_1 + 2m_2)  \\right.  \\right.\\nonumber\\\\\n&\\times& \\left.\\left. (A^{(2)}_2 + A^{(2)}_3 -A^{(1)}_1) - m'_1 (A^{(1)}_1 + A^{(1)}_2 -1) \\right) - A^{(1)}_1 + A^{(2)}_2 + A^{(2)}_3 - \\frac{1}{w'_{A}w''_{V}}(2A^{(3)}_1 + 2A^{(3)}_2 - 2A^{(2)}_1) \\right] \\nonumber\\\\\nf^a_5  (q^2)&=&   e N_{e'_1} \\frac{N_c}{16\\pi^3} \\int dx_2 d^2p'_{\\perp} \\frac{h'_{A}h''_{V}}{x_2\\hat N'_1 \\hat N''_1} (-4)\\cdot\\left [\n\\frac{1}{w'_{A}}\\left ( (m'_1 -m''_1) (A^{(2)}_3 - A^{(2)}_4) + (m'_1 + m''_1 - 2m_2)  \\right.  \\right.\\nonumber\\\\\n&\\times& \\left.\\left. (A^{(2)}_2 - A^{(2)}_3 ) + m'_1 (A^{(1)}_2 - A^{(1)}_1 ) \\right)  + A^{(2)}_2- A^{(2)}_3 - \\frac{1}{w'_{A}w''_{V}}( 2A^{(3)}_2 -2A^{(3)}_1) \\right]  \\ ,\n\\label{ffexpression}\n\\eeqs\n\n\\end{widetext}\nwhere the explicit expression of $A^{(i)}_j$ is given in Appendix\n\\ref{experssion}.\n\n\nFor the right-hand diagram in Fig.(\\ref{p1}), the amplitude ${\\cal A\n}^{\\mu\\nu\\rho}(b)$ can be obtained from ${\\cal A }^{\\mu\\nu\\rho}(a)$ by the\ninterchanges $m'_1 \\leftrightarrow m'_2$, $m''_1 \\leftrightarrow m''_2$, $m_2\n\\leftrightarrow m_1$, $N_{e'_1} \\leftrightarrow N_{e_2} $:\n\\beqs  i {\\cal A\n}^{\\mu\\nu\\rho}(b) &=& f^b_2 \\epsilon^{\\mu\\nu\\rho\\alpha}q_{\\alpha} +\nf^b_4 \\epsilon^{\\mu\\nu \\alpha \\beta}P^\\rho P_\\alpha q_\\beta  \\nonumber\\\\\n&+& f^b_5 \\epsilon^{\\rho\\mu \\alpha \\beta}P^\\nu P_\\alpha q_\\beta \\ .\n\\label{interchange_amplitude}\n\\eeqs\nThe coefficients $f_i$ in Eq.(\\ref{formfactor}) are the sum of contributions\nfrom two parts, ${\\cal A }^{\\mu\\nu\\rho}(a)$ and ${\\cal A }^{\\mu\\nu\\rho}(b)$:\n\\beq f_i(q^2) =f^a_i (q^2) + f^b_i (q^2) , \\quad (i=2, 4, 5) \\ .\n\\label{interchange_amp}\n\\eeq\nIn Eqs. (\\ref{ffexpression}) and (\\ref{interchange_amp}) we write these as\ngeneral form factors dependent on $q^2$, but note that in the physical $A \\to V\n\\gamma$ decay, $q^2=0$ for the real outgoing photon, so that these are simply\nconstant coefficients.  We use this generalization to nonzero $q^2$ because in\nthe light-front formalism, these form factors are calculated in the region\nwhere the photon momentum is not onshell, i.e., where $q^2 \\ne 0$.\nTo obtain the physical\nvalues $f_i(0)$ and calculate the decay rate, we take limit $q^2 \\to 0$.  This\nyields the resulting width\n\\begin{widetext}\n\\beqs\n\\Gamma(n{}^3P_1 \\to n'{}^3S_1) &=& \\frac{q^3}{24\\pi}\\left \\{ \\frac{2}{M''^2}[f^2_2 +4f_2f_4 M' q + 4 M'^2 q^2 f^2_4]  + \\frac{2}{M'^2}[ f^2_2 -4f_2 f_5  M' q+ 4 M'^2 q^2 f^2_5 ] \\right\\}   \\ ,\n\\label{widthLF}\n\\eeqs\n\\end{widetext}\nwhere $q= (M'^2-M''^2)\/(2M')$ is the momentum of the emitted photon.  In this\npaper we focus on E1 dipole transition rates, which are dominant, and hence\ndrop the subdominant $f_4$ and $f_5$ terms in the calculations.\n\n\n\\section{Wavefunctions for heavy quarkonium states}\n\\label{WF}\n\nThe wavefunctions $\\phi_{n's}$ and $\\phi_{np}$ can, in principle, be derived\nfrom relativistic light-front Bethe-Salpter type equations\n\\cite{Jaus:1989au,Cheung:1995ub}. However, as discussed in Refs.\n\\cite{Jaus:1989au} and \\cite{Isgur:1988gb}, there is a simpler approach, namely\nto use wavefunctions from nonrelativistic quark models with given potentials.\nAlthough a QCD-motivated potential has the form $V = -(4\/3)\\alpha_s(m_Q)\/r +\n\\sigma r$, as noted above, this involves the complication of requiring\nnumerical solutions of the Sch\\\"odinger equation.  To avoid this complication,\nRefs. \\cite{Jaus:1989au} and \\cite{Isgur:1988gb} used variational solutions of\nthe Schr\\\"odinger equation with a nonrelativistic harmonic oscillator\npotential. This approach was also adopted by\nRefs. \\cite{Jaus:1999zv,Cheng:2003sm,Choi:1997iq,Choi:1999nu}.  However, the\npredictions from this type of approach do not fit the measured widths for\n$\\Upsilon (nS)$ well, and to overcome this problem, modified harmonic\noscillator wavefunctions were suggested in \\cite{Ke:2010vn}. The normalization\nand explicit expressions for the modified harmonic wavefunctions are listed in\nAppendix \\ref{wavefunction}.\n\n\nIn the next section, we use the modified harmonic oscillator wavefunctions in\n\\cite{Ke:2010vn} to calculate numerically the radiative decay widths of\n$\\chi_{c1}(1P)$ and $\\chi_{b1}(nP)$ states and to compare these with\ntheoretical predictions from other models.\n\n\nSome comments are appropriate concerning approaches other than the light-front\napproach.  For the heavy quarkonium system, nonrelativistic potential models\nsuch as Cornell potential model have proved to be generally rather successful\nin fitting data\n\\cite{Eichten:1978tg,Eichten:1979ms,Eichten:1976jk,Eichten:1994gt,Buchmuller:1980su}. There\nare also analyses of relativistic corrections to potential models, such as\n\\cite{Gupta:1982kp,Moxhay:1983vu,Kwong:1988ae}. A relativistic quark model was\nproposed in Ref. \\cite{Godfrey:1985xj}. Screening effects were studied in\nRefs. \\cite{Laermann:1986pu,Chao:1992et,Ding:1993uy}, and additional potential\nmodels were used in \\cite{Sumino:2001eh,Recksiegel:2001xq}. In these potential\nmodels, the wavefunctions can be obtained by numerically solving the\nSchr\\\"odinger equations.  In future work it would be of interest to\ninvestigate the differences in radiative widths calculated using the\nphenomenological wavefunctions for the light-front quark model adopted\nhere (with modified harmonic oscillator wavefunctions) and wavefunctions from\npotential models. Here we focus on calculations using modified harmonic oscillator\nwavefunctions, and we compare these with results obtained from other\napproaches.\n\n\n\n\\section{Analysis of radiative transitions of $\\chi_{c1}(1P) $ and\n$\\chi_{b1}(nP)$ }\n\\label{ANALYSIS}\n\n\n\\begin{widetext}\n\n\\begin{table}\n\\begin{ruledtabular}\n\\caption{Decay width (in units of keV) of\n$\\chi_{c1}(1P) \\to J\/\\psi + \\gamma $ in the light-front approach, based on\nmodified harmonic oscillator wavefunctions \\cite{Ke:2010vn}.\nThe predictions from other models\n(relativistic quark model\\cite{Ebert:2002pp,Godfrey:2015dia},\nnonrelativistic screened potential model \\cite{Brambilla:2004wf})\nand experimental data from PDG \\cite{PDG} are also listed for comparison.\nThe parameters sets are CM1 and CM2. We use the PDG fitted value\n$\\Gamma_{\\chi_{c1}}=840 \\pm 40$ keV and $BR(\\chi_{c1}(1P) \\to J\/\\psi+\\gamma) =\n33.9 \\pm 1.2$ \\% \\cite{PDG}. For the entry referring to Ref.\n\\cite{Ebert:2002pp}, we list three values presented there, based on the\nspecific models used in that work.} \\label{tabCM}\n\\begin{tabular}{ccccccc}\n  Decay mode & CM1 & CM2 & exp.(PDG)\\cite{PDG}  & \\cite{Brambilla:2004wf}(NR)  & \\cite{Ebert:2002pp}  \\\\\n  \\hline\n$\\chi_{c1}(1P) \\to J\/\\psi+\\gamma$ & $324 \\pm 20$ & $282 \\pm 35$&\n$285 \\pm 30$ & 241  & 265\/285\/305  \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\n\\begin{table}\n\\begin{ruledtabular}\n\\caption{Coefficients $f_2$, $f_4$, and $f_5$ for\n$\\chi_{b1}(nP) \\to \\Upsilon (n'S) \\gamma$ in covariant light-front approach,\nwhere $f_i \\equiv f_i(q^2=0)$.}    \\label{tabFF}\n\\begin{tabular}{ccccccc}\n \n  Decay mode &  $f_2$  & $f_4 (\\text{GeV}^{-2})$ & $f_5 (\\text{GeV}^{-2})$  \\\\\n  \\hline\n   $\\chi_{b1}(1P) \\to \\Upsilon (1S) \\gamma$ &  $-0.94 \\pm0.06 $ & $0.0049 \\pm 0.0004$ &$-0.0083\\pm0.0002$ \\\\\n   $\\chi_{b1}(2P) \\to \\Upsilon (1S) \\gamma$ & $ +0.21 \\pm0.05$ &  $0.0019\\pm0.0006$ &$0.0037  \\pm0.0002$ \\\\\n   $\\chi_{b1}(2P) \\to \\Upsilon (2S) \\gamma$ &  $- 1.26\\pm0.10$ &  $0.0094\\pm0.0010$ &$-0.0071 \\pm0.0008$ \\\\\n   $\\chi_{b1}(3P) \\to \\Upsilon (1S) \\gamma$ & $-0.11\\pm 0.03$  &  $-0.0014\\pm0.0002$ &$-0.0021\\pm0.0003$ \\\\\n   $\\chi_{b1}(3P) \\to \\Upsilon (2S) \\gamma$ & $+0.29 \\pm 0.10 $ & $0.0038\\pm0.0016$ &$0.0050  \\pm0.0002$ \\\\\n   $\\chi_{b1}(3P) \\to \\Upsilon (3S) \\gamma$ &  $-1.39\\pm 0.06 $ &$0.0056\\pm0.0032$ &$-0.0087  \\pm0.0015$ \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\n\n\n\\begin{table}\n\\begin{ruledtabular}\n\\caption{Decay widths (in units of keV) of\n$\\chi_{b1}(nP) \\to \\Upsilon(n'S) + \\gamma$  E1 decays in the light-front\napproach, denoted $\\Gamma_{\\text{MSHO}}$,\nbased on modified simple\nharmonic oscillator (MSHO) wavefunctions \\cite{Ke:2010vn}. The predictions from\nother models (relativistic quark model\\cite{Ebert:2002pp,Godfrey:2015dia},\nnon-relativistic screened potential model \\cite{Li:2009nr}, and\nnonrelativistic constituent quark model \\cite{Segovia:2016xqb}) are also\nlisted for comparison, where  \\cite{Li:2009nr}$_0$ denotes results from\nthe $SNR_0$ (screened nonrelativistic) model and \\cite{Li:2009nr}$_1$ denotes results from the\n$SNR_1$ model. We also list the ratio $\\Gamma_{\\text{MSHO}}\/\\Gamma_{\\text {th(ave.)}}$, where $\\Gamma_{\\text {th(ave.)}}$ is average value of widths from other theoretical models.  }  \\label{tabM}\n\\begin{tabular}{ccccccccc}\n \n  Decay mode & $\\Gamma_{\\text{MSHO}}$   & \\cite{Ebert:2002pp}  & \\cite{Li:2009nr}$_0$  & \\cite{Li:2009nr}$_1$ & \\cite{Godfrey:2015dia} & \\cite{Segovia:2016xqb}& $\\Gamma_{\\text{MSHO}}\/\\Gamma_{\\text {th(ave.)}}$ \\\\\n  \\hline\n   $\\chi_{b1}(1P) \\to \\Upsilon (1S) \\gamma$ &  $37.3 \\pm 4.8 $ &  36.6 &33.6 &30.0 &29.5 &35.66& $1.12\\pm 0.15$\\\\\n   $\\chi_{b1}(2P) \\to \\Upsilon (1S) \\gamma$ &  $10.6 \\pm 5.5 $   &7.49 &12.4&8.56 &5.5 &9.13& $1.23\\pm 0.64$\\\\\n   $\\chi_{b1}(2P) \\to \\Upsilon (2S) \\gamma$ &  $10.0 \\pm 1.7 $   &14.7 &15.9&13.8&13.3 &15.89& $0.68\\pm 0.12$\\\\\n   $\\chi_{b1}(3P) \\to \\Upsilon (1S) \\gamma$ &  $6.1 \\pm 3.9 $  & &6.80&3.39 &1.3 &4.17 & $1.56\\pm 1.00$\\\\\n   $\\chi_{b1}(3P) \\to \\Upsilon (2S) \\gamma$ &  $4.7 \\pm 3.2$   & & 5.48&5.39 &3.1 &4.58& $1.01\\pm 0.69$ \\\\\n   $\\chi_{b1}(3P) \\to \\Upsilon (3S) \\gamma$ &  $3.6 \\pm 0.4$   &  &12.0&9.97 &8.4&9.62& $0.36\\pm 0.04$ \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\n\n\\end{widetext}\n\nIn this section we apply the light-front formalism for the decay $A (1^{++} )\n\\to V (1^{--}) + \\gamma $ to the analysis of the radiative decays\n$\\chi_{c1}(1P) \\to J\/\\psi + \\gamma$ and $\\chi_{b1}(nP) \\to \\Upsilon\n(n'S)+\\gamma$. We present the results of our numerical calculations of form\nfactors (evaluated at $q^2=0$) and decay widths. For the charmonium\n$\\chi_{c1}(1P)$ decay, we compare our result with experimental data on the\nwidth, as listed in the Particle Data Group Review of Particle Properties (RPP)\n\\cite{PDG}. Although the RPP lists this width for the decay $\\chi_{c1}(1P) \\to\nJ\/\\psi$, it does not list widths for the $\\chi_{b1}(nP) \\to \\Upsilon (n'S) +\n\\gamma$ decays, only branching ratios.  Since our calculation yields the width\nitself, and a calculation of the branching ratio requires division by the total\nwidth in each case, we therefore compare our results on the branching ratios\nfor these decays with predictions from other models, including the relativistic\nquark model \\cite{Ebert:2002pp,Godfrey:2015dia}, the non-relativistic screened\npotential model \\cite{Li:2009nr}, and the nonrelativistic constituent quark\nmodel \\cite{Segovia:2016xqb}. For each decay, we have performed numerical\ncalculations based on modified harmonic oscillator wavefunctions as discussed\nin \\cite{Ke:2010vn}.\n\nFirst, we study the charmonium radiative decay $\\chi_{c1}(1P) \\to J\/\\psi\n+\\gamma$.  The parameter sets that we use are as follows, with labels\nindicated:\n\n\\begin{enumerate}\n\n\\item CM1: $m_c$ = 1.4 \\ GeV, \\\\\n $\\beta_{\\chi_{C1}(J\/\\psi)}$ = 0.639$\\pm 0.020$ \\ GeV.\n\n\\item CM2:  $m_c$ = 1.5 \\ GeV, \\\\\n  $\\beta_{\\chi_{C1}(J\/\\psi)}$ = 0.600$\\pm 0.020$ \\ GeV.\n\n\\end{enumerate}\n\nWe present our results in Table \\ref{tabCM}, with the uncertainties arising\nfrom the uncertainties in the $\\beta$ parameters, as in \\cite{Ke:2013zs}.\nAs one can see from Table \\ref{tabCM}, our results agree with experimental\ndata within the range of experimental and theoretical uncertainties. The\ntheoretical uncertainties arise from the value of $m_c$ taken and also from the\nmodel used. The model-dependent uncertainties will be evident from our\ncomparison of predictions from various models.\n\n\nNext, we proceed to analyze the radiative decays of P-wave $b \\bar b$\nstates. We use the modified harmonic oscillator wavefunctions, which have been\nsuccessfully applied to the study of radiative decays of $\\Upsilon (nS) \\to\n\\eta_b + \\gamma$ \\cite{Ke:2010vn}.  In this case, the LFQM has the following\nparameters: the mass of the quark, $m_b$, the harmonic oscillator wavefunction\nparameter for $\\chi_{b1}(nP)$, $\\beta_{\\chi_{b1(nP)}}$, and the wavefunction\nparameter for $\\Upsilon(nS)$ $\\beta_{\\Upsilon(nS)}$.  For the mass of the\nquark, we use $m_b = 4.8$ GeV. This is an effective $b$-quark mass chosen to\noptimize the fit to these radiative transitions, as has been done in a number\nof other studies; for example, the recent comprehensive study\n\\cite{Godfrey:2015dia} uses the value $m_b=4.977$ GeV.\n\n\nFor the effective harmonic oscillator wavefunction parameters, there are two\nchoices. One is to use a single parameter $\\beta$ for all states in the $b \\bar\nb$ system. In this case, the wavefunctions correspond to eigenstates of the\nharmonic oscillator Hamilton with $V \\propto r^2$, and hence the energy\nsplitting between different energy levels is \\cite{quiggrosner79}\n$\\Delta E \\propto \\mu^{-1\/2}$, where $\\mu=m_Q\/2$ is the reduced mass of $Q\\bar\nQ$ system. This does not account for the observed approximate equality of mass\nsplittings $m(\\psi(2S))-m(J\/\\psi) \\simeq m(\\Upsilon(2S))-m(\\Upsilon(1S))$.\nTherefore, a more practical choice\nis to treat $\\beta$ as variational parameter to fit each state separately. For\nexample, in Ref. \\cite{Godfrey:2015dia}, the authors obtain $\\beta$ by equating\nthe rms radius of the harmonic oscillator wavefunction for the specified states\nwith the rms radius of the wavefunctions calculated using the relativized quark\nmodel. Explicitly, for $n=1$, $\\beta \\sim 0.9 -1.2$ GeV, for $n=2$,\n$\\beta \\sim 0.7 -0.8$ GeV, and for $n=3$, $\\beta \\sim 0.6 -0.7$ GeV.\nFor our modified harmonic wavefunctions, these results are not exact,\nbut can serve as an estimate of the range of wavefunction parameters. In our\nanalysis, we use the following values of wavefunction parameters:\n\n\\begin{enumerate}\n\n\\item\n$\\beta_{\\chi_{b1}(1P)}$ = 1.00$\\pm0.02$ GeV,\n\n\\item\n$\\beta_{\\chi_{b1}(2P)}$ =  0.71$\\pm0.02$ GeV,\n\n\\item\n$\\beta_{\\chi_{b1}(3P)}$ = 0.70$\\pm0.02$ GeV,\n\n\\item\n$\\beta_{\\Upsilon(1S)}$ = 0.90$\\pm0.02$ GeV,\n\n\\item\n$\\beta_{\\Upsilon(2S)}$ = 0.71$\\pm0.02$ GeV,\n\n\\item\n\n$\\beta_{\\Upsilon(3S)}$ = 0.70$\\pm0.02$ GeV.\n\n\\end{enumerate}\n\nHere we have used estimated values of the uncertainties in these\nparameters corresponding to those that we used in our previous study\n\\cite{Ke:2013zs}.  The uncertainties that we include with our resultant\ncalculations of radiative decay widths incorporate these uncertainties.\n\n\nFor the values of form factors, we show typical results in Table {\\ref{tabFF}.\n  The numerical results for the decay widths calculated with our parameter\n  setting are listed in Table {\\ref{tabM}.  In both of these tables, we include\n    the estimated uncertainties arising from the uncertainties in the input\n    value of $m_b$ and the input values of the $\\beta$ parameters. Since for\n    $\\chi_{b1}(nP)$ system, only branching ratios are experimentally\n    determined, we compare our results, denoted $\\Gamma_{\\text {MSHO}}$, with\n    those from other theoretical models. As an rough estimation, we define the\n    average values of widths from these theoretical models\n    \\cite{Ebert:2002pp,Godfrey:2015dia,Li:2009nr,Segovia:2016xqb} to be\n    $\\Gamma_{\\text{th(ave.)}}$.  It should be noted that for many of the decay\n    modes, there is a substantial spread of values of branching ratios\n    predicted by different models. We then calculate the ratio $\\Gamma_{\\text\n      {MSHO}}\/\\Gamma_{\\text{th(ave.)}}$ and list this ratio in Table\n    {\\ref{tabFF}. The decay $\\chi_{b1}(1P) \\to \\Upsilon(1S) + \\gamma$ has a\n      measured branching ratio $BR(\\chi_{b1}(1P) \\to \\Upsilon(1S) + \\gamma) =\n      33.9 \\pm 2.2$ \\% \\cite{PDG}.  For this decay mode, our predicted width\n      agrees well with the average of the other models and, furthermore, the\n      predictions of these other models agree well among themselves.  The\n      measured branching ratios for the radiative decays of the $\\chi_{b1}(2P)$\n      are $BR(\\chi_{b1}(2P) \\to \\Upsilon(2S) + \\gamma) = 19.9 \\pm 1.9$ \\% and\n      $BR(\\chi_{b1}(2P) \\to \\Upsilon(1S) + \\gamma) = 9.2 \\pm 0.8$ \\%\n      \\cite{PDG}. Our predicted width for the first of these decays is in good\n      agreement with the average of the predictions of other models, while our\n      predicted width for the second of these decays is slightly smaller than\n      this average. $\\chi_{bJ}(3P)$ states have recently been observed at the\n      LHC via their radiative decays \\cite{Aad:2011ih,Aaij:2014hla} (although\n      no branching ratios for these decays are listed yet by the PDG). For\n      radiative decays of $\\chi_{b1}(3P)$ to $\\Upsilon(1S)$ and $\\Upsilon(2S)$,\n      our LFQM predictions are in good agreement, to within uncertainties, with\n      other models, while our prediction for the decay to $\\Upsilon(3S)$ is\n      somewhat smaller than the predictions from other models.\n\n      In general, these results show that the light-front quark model with\n      phenomenological meson wavefunctions (specifically, modified harmonic\n      oscillator wavefunctions), is suitable for the calculation of $nP \\to\n      n'S$ radiative decay widths, since this model gives reasonable\n      predictions for these widths, as compared with experimental data and\n      other theoretical approaches.  The results from the calculations in the\n      covariant light-front approach and corresponding\n      nonrelativistic\/relativized quark model calculations reflect some\n      differences in the predictions of decay widths, which are related to\n      differences in the properties of these respective models.  Specifically,\n      nonrelativistic\/relativized quark models contain different ways of\n      including relativistic corrections and also truncations of these\n      relativistic effects, while in the LFQM these relativistic corrections\n      are systematically included.  This shows one advantage of the covariant\n      light-front approach, namely, that it is a fully relativistic formalism,\n      and one does not need to carry out a reduction from relativistic\n      interaction terms to the nonrelativistic limit.\n\n\nOne drawback in the current LFQM is that we do not know the exact form of the\nlight-front wavefunctions and hence only use trial wavefunctions. This\nproblem is more serious for excited states, because for excited $b \\bar b$\nstates with radial quantum numbers $n \\ge 2$, where $\\Lambda_{QCD}$ is larger\nthan the typical binding energy of the state, the Coulombic type potential is\nno longer a very good approximation \\cite{Brambilla:2010cs}, so we have larger\nuncertainties in the $b \\bar b$ wavefunction that serves as input in\nlight-front quark model.  This can be seen from Table \\ref{tabM}; for reasonable parameters,\nthe decay width of $\\chi_{b1}(1P)$ from the LFQM agrees well with predictions\n from nonrelativistic\/relativized quark\nmodels, but for excited states, the LFQM calculations for two channels do\nnot match perfectly with predictions from these nonrelativistic\/relativized\nquark models.  As been pointed out in Ref. \\cite{Brambilla:2010cs}, for\nradiative transition of these excited $b \\bar b$ states, we rely on\nphenomenological models, but these do not always agree with QCD in the\nperturbative regime. Even though the LFQM is a fully relativistic approach,\nthere is thus motivation for further theoretical work to gain a better\nunderstanding of the determination of light-front wavefunctions for\n$Q \\bar Q$ states.\n\n\n\\section{Conclusion}\n\\label{CON}\n\nIn this paper we have derived formulas for the radiative decay of $1^{++}$\nheavy mesons via the channel $1^{++} \\to 1^{--} +\\gamma$ in the light-front\nquark model. Then we have applied these to calculate the coefficients $f_i$ and\nthe radiative decay widths of $\\chi_{c1}(1P)$ and $\\chi_{b1}(nP)$ via the\nrespective channels $\\chi_{c1}(1P) \\to J\/\\psi + \\gamma$ and\n$\\chi_{b1}(nP) \\to \\Upsilon(n'S) + \\gamma$. Within the LFQM framework, we have\nadopted modified harmonic-oscillator wavefunctions.\nWe have shown that most of the predictions of the LFQM with modified\nharmonic-oscillator wavefunctions are in reasonable agreement with data and\nother model calculations.\n\n\n\\begin{acknowledgments}\n  We are grateful to Prof. Robert Shrock for his helpful suggestions and\n  assistance. This research was partially supported by the NSF grant\n  NSF-PHY-13-16617. We are also grateful to Profs. Hong-Wei Ke and Xue-Qian Li\n  for collaboration on our previous related work \\cite{Ke:2013zs}.\n\\end{acknowledgments}\n\n\n\n\\begin{appendix}\n\n\\section{ Time reversal and Parity transformations of  amplitude}\n\\label{PT invariance}\n\n\\subsection{Time Reversal Transformation}\n\nThe action of time reversal on on S-matrix element $\\langle \\beta | H | \\alpha\n\\rangle $ is defined to be $(\\langle \\tilde\\beta | {\\cal T}H {\\cal T}^{-1}|\n\\tilde \\alpha \\rangle)^{*}$, where $|\\tilde \\alpha \\rangle={\\cal T} | \\alpha\n\\rangle $. So the time-reversal invariance of electromagnetic interaction is\n\\cite{Dudek:2006ej}:\n\\beq\n\\langle \\beta | H_{\\text em} | \\alpha \\rangle =(\\langle \\tilde\\beta | {\\cal T}H_{\\text em} {\\cal T}^{-1}| \\tilde \\alpha \\rangle)^{*}=(\\langle \\tilde\\beta | H_{\\text em} | \\tilde \\alpha \\rangle)^{*}\n\\label{Ttrans}\n\\eeq\nwhere we use the time-reversal invariance of the electromagnetic Hamiltonian\noperator: $H_{\\text em}={\\cal T}H_{\\text em} {\\cal T}^{-1}$.\n\nFor a state with 3-momentum $\\vec{p}$, spin $J$ and $z$-component of spin $m$,\nthe time-reversal transformation is ${\\cal T} |\\vec{p},J,m\\rangle=\n\\zeta(-1)^{J-m}|-\\vec{p},J,-m\\rangle $ (for vector and axial-vector states,\n$\\zeta =+1$). After contractions with the associated field operator, this\namounts to the change of polarization: $\\varepsilon^{\\mu}(\\vec{p},m) \\to\n\\zeta(-1)^{J-m}\\varepsilon^{\\mu}(-\\vec{p},-m) =\\zeta(-1)^{J+1} {\\text\n  {P}}^{\\mu}_{\\nu}\\varepsilon^{\\nu*}(\\vec{p},m) $, where we have used the\nrelation $\\varepsilon^{\\mu*}(\\vec{p},m)=(-1)^{m+1}{\\text\n  {P}}^{\\mu}_{\\nu}\\varepsilon^{\\nu}(-\\vec{p},-m)$, and ${\\text {P}}^{\\mu}_{\\nu}\n= \\text {diag} (1,-1,-1,-1)$ represents spatial inversion \\cite{Dudek:2006ej}.\n\nThe general amplitude in Eq.(\\ref{formfactor}) should satisfy the time-reversal\ninvariance condition in Eq.\\ref{Ttrans}. Let us consider the $f_2$ term\nfirst. Without loss of generality, we can choose polarization (+,+,+) states;\nthen this is given by\n\\beq\n{\\cal M}_{+++}=-i\\varepsilon^{*}_{\\mu}(q,+)\n\\varepsilon'_{\\nu}(P',+)\\varepsilon''^{*}_{\\rho}(P'',+)f_2\n\\epsilon^{\\mu\\nu\\rho\\alpha}q_{\\alpha} \\ .\n\\label{f2M+++}\n\\eeq\nIn this case where the three polarization vectors are all transversal and only\ncarry spatial components of Lorentz indices, the index of the photon momentum\n$q_{\\alpha}$ has to be $\\alpha=0$:\n\\beq\n{\\cal M}_{+++}=-i\\varepsilon^{*}_{i}(q,+)\n\\varepsilon'_{j}(P',+)\\varepsilon''^{*}_{k}(P'',+)f_2 \\epsilon^{ijk}q_{0} \\ .\n\\label{f2M+++2}\n\\eeq\nUnder a time-reversal transformation, $q^{0} \\to q^{0}$, $\\epsilon^{i}\n(\\epsilon'^{i}, \\epsilon''^{i})$ $\\to$ $(-1)\\epsilon^{i*}(\\epsilon'^{i*},\n\\epsilon''^{i*} )$ and\n\\beqs\n{\\widetilde {\\cal M}}_{+++}&=& -(-1)^3\\left (i\\varepsilon_{i}(q,+) \\varepsilon'^{*}_{j}(P',+)\\varepsilon''_{k}(P'',+)f_2 \\epsilon^{ijk}q_{0} \\right)^{*} \\nonumber\\\\\n&=&  -i\\varepsilon^{*}_{i}(q,+) \\varepsilon'_{j}(P',+)\\varepsilon''^{*}_{k}(P'',+)f^{*}_2 \\epsilon^{ijk}q_{0}  \\ .\n\\eeqs\nAccording to Eq. (\\ref{Ttrans}), the amplitude is time-reversal invariant if\n\\beq\n{\\cal M}_{+++} ={\\widetilde {\\cal M}}_{+++} \\ \\to \\  f_2 = f^*_2 \\ ,\n\\eeq\nwhich is satisfied as we can see from the explicit expression of $f_2$ in\nEq.(\\ref{ffexpression}). Using an equivalent analysis, we can prove that\nthe $f_4$ and $f_5$ terms also preserve time-reversal invariance.\n\n\\subsection{Parity Transformation}\n\nFor a physical state $| \\alpha \\rangle$, the action of a parity transformation\nis ${\\cal P} | \\alpha \\rangle =| \\alpha' \\rangle$.  The parity\ninvariance of the electromagnetic interaction is expressed as\n\\beq\n\\langle \\beta| H_{\\text {em}} | \\alpha \\rangle = \\langle \\beta|{\\cal P}^{-1} {\\cal P} H_{\\text {em}} {\\cal P}^{-1}{\\cal P} | \\alpha \\rangle= \\langle \\beta'| H_{\\text {em}} | \\alpha' \\rangle\n\\label{Ptrans}\n\\eeq\nThe parity transformation of a state with 3-momentum $\\vec{p}$, spin $J$, and\n$z$-component of spin $m$ is defined as ${\\cal P}|\\vec{p},J,m\\rangle=\n\\eta_{P}|-\\vec{p},J,m\\rangle $, where $\\eta_{P}$ is the intrinsic parity of\nthis state. For a vector meson, $\\eta_{P}=-1$, and for an axial vector meson,\n$\\eta_{P}=+1$. After contractions with the associated field operator, this\namounts to the change of polarization: $\\varepsilon^{\\mu}(\\vec{p},m) \\to\n\\eta_{P}\\varepsilon^{\\mu}(-\\vec{p},m) =-\\eta_{P}{\\text\n  {P}}^{\\mu}_{\\nu}\\varepsilon^{\\nu}(\\vec{p},m) $, where we have used the\nrelation $\\varepsilon^{\\mu}(-\\vec{p},m) = -{\\text\n  {P}}^{\\mu}_{\\nu}\\varepsilon^{\\nu}(\\vec{p},m) $.\n\nThe general amplitude in Eq.(\\ref{formfactor}) should satisfy the parity\ninvariance condition in Eq.\\ref{Ptrans}. We take the $f_2$ term as an example\nto demonstrate this requirement. Without loss of generality, we can choose\nthe polarization (+,+,+) states, for which the amplitudes are given by\nEq.(\\ref{f2M+++}) and Eq.(\\ref{f2M+++2}).\n\nUnder a parity transformation, $q^{0}\\to q^{0}$, $\\epsilon^{i} (\\epsilon'^{i},\n\\epsilon''^{i})$ $\\to$ $\\eta_{P}(+1)\\epsilon^{i}(\\epsilon'^{i}, \\epsilon''^{i}\n)$, the amplitude ${\\cal M}_{+++}$ is transformed to\n\\beqs\n{\\cal M'}_{+++}&=&-\\eta_{V}\\eta_{A}\\eta_{\\gamma}i\\varepsilon^{*}_{i}(q,+) \\varepsilon'_{j}(P',+)\\varepsilon''^{*}_{k}(P'',+)f_2 \\epsilon^{ijk}q_{0} \\nonumber\\\\\n&=&-(+1)i\\varepsilon^{*}_{i}(q,+) \\varepsilon'_{j}(P',+)\\varepsilon''^{*}_{k}(P'',+)f_2 \\epsilon^{ijk}q_{0} \\ , \\nonumber\\\\\n\\label{M+++P}\n\\eeqs\nwhere the intrinsic parities of $V$, $A$ and $\\gamma$ are\n$\\eta_{V}=-1$, $\\eta_{A}=+1$ and $\\eta_{\\gamma}=-1$, respectively. From\nEq.(\\ref{M+++P}), we can see ${\\cal M'}_{+++}={\\cal M}_{+++}$; hence parity is\nconserved for the $f_2$ term. Applying the same method of analysis, we can\nprove that the $f_4$ and $f_5$ terms also preserve parity invariance.\n\n\n\n\\section{The wavefunctions}\n\\label{wavefunction}\n\nThe normalization of the S-wave meson wavefunction in the light-front framework\nis\n\\beq\n\\frac{1}{2(2\\pi)^3} \\int dx_2 dp^2_{\\perp} |\\phi_{n's}(x_2, p_{\\perp})|^2 =1.\n\\eeq\nHere $\\phi_{n's}(x_2, p_{\\perp})$ is related to the wavefunction in normal\ncoordinates $\\psi_{n's}(p)$ by\n\\beq\n\\phi_{n's} (x_2, p_{\\perp})= 4\\pi^{\\frac{3}{2}} \\sqrt{\\frac{d p_z}{dx_2}}\\psi_{n's} (p) \\ , \\quad \\frac{dp_z}{dx_2}= \\frac{e'_1e_2}{x_1x_2 M'_{0}} \\\n\\eeq\nThe normalization of $\\psi_{n's}(p)$ is given by\n\\beq\n\\int d{\\bf  p}^3 |\\psi_{n's}(p)|^2 = 4\\pi  \\int  p^2 dp |\\psi_{n's}(p)|^2 =1 \\ .\n\\eeq\nThe normalization for the P-wave meson wavefunction in the light-front\nframework is \\cite{Cheng:2003sm}\n\\beq\n\\frac{1}{2(2\\pi)^3} \\int dx_2 dp^2_{\\perp} |\\phi_{np}(x_2, p_{\\perp})|^2 p_i p^{*}_j=\\delta_{ij} \\ ,\n\\label{normalP}\n\\eeq\nwhere $p_i=(p^{+}, p^{-},p_z)$. In terms of the P-wave wavefunction in normal\ncoordinates,\n\\beq\n\\phi_{np} (x_2, p_{\\perp})= 4\\pi^{\\frac{3}{2}} \\sqrt{\\frac{d p_z}{dx_2}}\\psi_p (p) \\ , \\quad \\frac{dp_z}{dx_2}= \\frac{e'_1e_2}{x_1x_2 M'_{0}} \\\n\\eeq\nwe have the following normalization condition:\n\\beq\n\\frac{1}{3} \\cdot 4\\pi  \\int^{\\infty}_{0}|\\psi_{np}(p)|^2 p^4 dp = 1 \\ .\n\\eeq\n\nFor the gaussian type 1P and 1S wavefunctions, we have the relation\n\\beq\n\\psi^{1P}_{p}(p)=\\sqrt{\\frac{2}{\\beta^2}} \\psi^{1S}_s (p) \\ .\n\\eeq\n\nThe modified harmonic oscillator n-S wavefunctions in the light-front\napproach are \\cite{Ke:2010vn}\n\\beqs\n\\psi^{1S}_{s,M}(p) &=& \\left( \\frac{1}{\\beta^2 \\pi}  \\right)^{\\frac{3}{4}} \\exp \\left(  -\\frac{1}{2}\\frac{p^2}{\\beta^2} \\right) \\nonumber \\\\\n\\psi^{2S}_{s,M}(p) &=& \\left( \\frac{1}{\\beta^2 \\pi}  \\right)^{\\frac{3}{4}} \\exp \\left(  -\\frac{2^\\delta}{2}\\frac{p^2}{\\beta^2} \\right) \\left( a'_2 -b'_2 \\frac{p^2}{\\beta^2} \\right) \\nonumber \\\\\n \\psi^{3S}_{s,M}(p) &=& \\left( \\frac{1}{\\beta^2 \\pi}  \\right)^{\\frac{3}{4}} \\exp \\left(  -\\frac{3^\\delta}{2}\\frac{p^2}{\\beta^2} \\right) \\left( a'_3 -b'_3 \\frac{p^2}{\\beta^2} + c'_3\n  \\frac{p^4}{\\beta^4} \\right) \\nonumber \\\\\n\\eeqs\nwhere\n\\beqs\n&& a'_2 = 1.88684 \\quad b'_2 =1.54943   \\nonumber \\\\\n&& a'_3 = 2.53764 \\quad b'_3 =5.67431 \\quad c'_3 = 1.85652 \\nonumber \\\\\n&& \\delta =1\/1.82 \\ .\n\\eeqs\nThe $n$P wavefunction, is related to $n$S wavefunction as follows:\n\\beq\n\\psi^{nP}_{p,M}(p)=\\frac{C_n}{\\beta}\\psi^{nS}_{s,M}(p) \\ ,\n\\eeq\nwhere the constants $C_n$ can be determined by the normalization condition in\nEq.(\\ref{normalP}) as\n\\beqs\n&&C_1 =\\sqrt{2}=1.41421\\nonumber\\\\\n&&C_2 =1.23833 \\nonumber\\\\\n&&C_3 =1.13215 \\ .\n\\eeqs\n\n\\section{Some expressions in the light-front formalism}\n\\label{experssion}\n\n\nIn the covariant light-front formalism we have\n\\beqs\nM'^2_0 &=&(e'_1 + e_2)^2 = \\frac{p'^2_{\\perp}+m'^2_1}{x_1}+ \\frac{p'^2_{\\perp}+m^2_2}{x_2} \\nonumber\\\\\nM''^2_0 &=&(e''_1 + e_2)^2 = \\frac{p''^2_{\\perp}+m''^2_1}{x_1}+ \\frac{p''^2_{\\perp}+m^2_2}{x_2}  \\nonumber\\\\\n\\tilde M'_{0} &=&  \\sqrt{M'^2_0-(m'_1-m_2)^2}\\nonumber\\\\\n\\tilde M''_{0} &=& \\sqrt{M''^2_0-(m''_1-m_2)^2} \\nonumber\\\\\np'_z &=& \\frac{x_2 M'_{0}}{2}- \\frac{m^2_2+p'^2_{\\perp}}{2x_2 M'_{0}} \\nonumber\\\\\np''_z &=& \\frac{x_2 M''_{0}}{2}- \\frac{m^2_2+p''^2_{\\perp}}{2x_2 M''_{0}} \\nonumber\\\\\ne'_1 &=&\\sqrt{m'^2_1+p'^2_{\\perp}+p'^2_{z}} \\nonumber \\\\\ne''_1 &=&\\sqrt{m''^2_1+p''^2_{\\perp}+p''^2_{z}} \\nonumber \\\\\ne_2 &=&\\sqrt{m^2_2+p'^2_{\\perp}+p'^2_{z}} \\ ,\n\\eeqs\n\nThe explicit expressions for $A^{(i)}_{j} (i,j=1 \\sim 4) $ are\n\\beqs\n&&A^{(1)}_1=\\frac{x_1}{2} \\ , \\ A^{(1)}_2= A^{(1)}_1- \\frac{p'_{\\perp}\\cdot q_{\\perp}}{q^2}\\ , \\nonumber\\\\\n&&A^{(2)}_{1}=-p'^2_{\\perp}-\\frac{(p'_{\\perp}\\cdot q_{\\perp})^2}{q^2}, \\ A^{(2)}_2= (A^{(1)}_1)^2, \\nonumber\\\\\n&& A^{(2)}_3= A^{(1)}_1 A^{(1)}_2 \\ , \\   A^{(2)}_4= (A^{(1)}_2)^2-\\frac{1}{q^2}A^{(2)}_1 \\ ,\\nonumber\\\\\n&& A^{(3)}_1 = A^{(1)}_1 A^{(2)}_1 \\ , A^{(3)}_2 = A^{(1)}_2 A^{(2)}_1 \\ , \\nonumber\\\\\n&& A^{(3)}_3 = A^{(1)}_1 A^{(2)}_2 \\ , A^{(3)}_4 = A^{(1)}_2 A^{(2)}_2 \\  .\n\\eeqs\n\n\n\\end{appendix}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:intro}\n\nA foundation model~\\cite{bommasani2021opportunities} is usually a big model trained on broad data (generally using self-supervision at scale) that can be fine-tuned to a wide range of downstream tasks and has aroused extensive attention due to its impressive quality improvements and emergent capabilities~\\cite{brown2020language,radford2021learning,raffel2020exploring,liu2019roberta}.\nIn speech community, self-supervised pre-training speech foundation models on a large amount of unsupervised speech has shown impressive quality improvements on various speech recognition tasks \\cite{zhang2022bigssl,hwang2022large}. There are two main categories of speech self-supervised learning algorithms. One direction is to reconstruct (APC~\\cite{chung2020generative}, MPC~\\cite{wang2020unsupervised}) or predict (Wav2vec~\\cite{oord2018representation,Schneider2019_wav2vec,huo2021incremental}) the input feature directly. The other direction is building a BERT-style self-supervised learning model by bridging the gap between continuous speech signal and discrete text tokens,\nsuch as Wav2vec 2.0~\\cite{baevski2020wav2vec}, HuBERT~\\cite{hsu2021hubert}, w2v-BERT~\\cite{chung2021w2v}  and BEST-RQ~\\cite{chiu2022self}. After pre-training the speech foundation model using the self-supervised loss, we initialize the encoder of the downstream task using the pre-trained model and fine-tune it on the supervised data.  \n\nA large general-purpose foundation model with millions or even billions of parameters can be adapted to many downstream tasks. However, it is challenging to perform  separate adaptations for many tasks efficiently with only a small amount of supervised data each task. There have been existing works investigating to reduce the number of parameters required for fine-tuning the foundation model. BitFit~\\cite{devlin2018bert} proposes a sparse-finetuning method where only the bias terms of the foundation model are updated.  Houlsby et al.~\\cite{houlsby2019parameter} propose to insert Adapter modules between the layers in the fixed pre-trained model and each  module is a small trainable feed-forward neural network. Other works~\\cite{karimi2021compacter,hu2021lora} reduce the number of parameters further by exploiting a low-rank approximation of the Adapter. Although these  parameter-efficient methods achieve decent performance on the downstream task with a significant reduction in the trainable parameters, their required computational memory cost and training time are still very high because of the following two reasons: 1) using the output of the highest layer in the foundation model only for downstream tasks, which leads to the inefficiency of the feature usage and requires to update the foundation model to adapt it to the downstream tasks;  2) adding\/updating sparse parameters in the foundation model, which requires a full backpropagation process from the top to the bottom of the network to compute the gradients of the trainable parameters. Thus, a resource-efficient transfer learning method, which can achieve comparable performance with small number of trainable parameters, low computational memory cost and fast training speed, is required for efficient adaptation of the foundation model to many downstream tasks. \n\nRecently, Pasad et al.~\\cite{pasad2021layer} analyze the layer-wise features of a self-supervised (wav2vec2.0) pre-trained speech representation model and finds that the middle layers encode the most contextual and high-level information.  The bottom or top few layers, on the other hand, focus on the lower-level information and encode more local representations. Arunkumar et al.~\\cite{arunkumar2022investigation} investigate the ensemble features of self-supervised pre-trained models for\nASR and finds that features from different self-supervised learning methods are complementary and the ensemble of features is beneficial for the downstream speech recognition tasks. Although behaviors of the layer-wise features and features from multiple self-supervised pre-trained models are explored, neither of them consider the resource efficiency in the fine-tuning stage and there is no investigation about the feature fusion of layer-wise features from a single pre-trained model on downstream tasks. \n\nIn this paper, we propose a novel resource-efficient transfer learning method for speech foundation models. specifically, we treat the foundation model as a frozen feature extractor and fuse the multi-level features from the foundation model hierarchically. We conduct extensive experiments to investigate different ways of feature fusion for the foundation model. Experimental results show that the proposed method can achieve better performance on the ASR task than existing parameter-efficient fine-tuning algorithms with fewer number of trainable parameters, less computational memory cost and faster training speed. After combining with Adapters at all layers, the proposed method can achieve the same performance as fine-tuning the whole model with $97\\%$ fewer trainable encoder parameters and $53\\%$ faster training speed.\n\n\\section{Related Works}\n\\label{sec:related}\n\n\n\\section{Experimental Setup}\n\\label{sec:experiments}\n\n\\subsection{Foundation Model And Task}\nThe foundation model used in the paper is a $2$-layer convolutional network followed by a $24$-layer conformer encoder with hidden dimension $1024$ and about $600$M parameters in total. Each conformer layer~\\cite{gulati2020conformer} is a convolution-augmented\ntransformer network, which consists of attention, feed-forward and convolutional modules. The model input is a vector of\nsize $128$ logMel features and SpecAugment~\\cite{Park2019} is also applied to increase model robustness. We pre-train the $600$M conformer encoder using the BEST-RQ \\cite{chiu2022self} algorithm for $800$K steps. For the downstream speech recognition task, we initialize the encoder using the pre-trained speech foundation model and the output of the encoder is used as input to an RNN-T~\\cite{zhang2022bigssl} along with a $6$-layer LSTM decoder and dimension $768$. We train with Adam optimizer for both pre-training and fine-tuning, and use exponential moving\naveraging (EMA) with decay rate 0.9999 for fine-tuning only. \nWe update the trainable encoder parameters and LSTM decoder which has $124$M trainable parameters on Voice Search data for $100$K steps. If not described explicitly, the parameter efficiency refers to the reduction of the trainable parameters in the encoder only. All experiments are performed on TPUs.\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{linear_wer_layers.png}\n    \\caption{Voice Search WER when extracting features from different layers of the foundation model.}\n    \\label{fig:linear_wer}\n\\end{figure}\n\n\n\\subsection{Training Data} \n\nWe use two sources of training data in this work. Following ~\\cite{zhang2022bigssl}, we collect $800$K hours unsupervised  English Youtube data and pre-train the $600$M foundation model on the randomly segmented audio-only Youtube speech using the BEST-RQ algorithm~\\cite{chiu2022self}.\nIn addition, the supervised English Voice Search (VS) data contains $5$K hours of labeled voice search audio~\\cite{Narayanan2018} and is used to fine-tune the conformer encoder and RNNT-T decoder for the ASR task. All data are collected and deidentified in accordance with Google AI principles~\\cite{aiprinciple}. \n\n\\subsection{Evaluation}\nIn this paper, we calculate the word error rate (WER) on the Voice Search (VS) test dataset to measure the quality of the model on the downstream speech recognition task. Apart from WER, we compare the number of trainable parameters, computational memory cost and training speed at the same time for resource efficiency. The target of this paper is to propose a method, which can achieve low WER with small number of trainable parameters, low computational memory cost and fast training speed. \n\n\\section{Linear Feature Fusion of The Foundation Model}\n\n\n\\label{sec:linear}\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{LinearFF_v2.png}\n    \\caption{Linear feature fusion from multiple layers of the foundation model, using a $6$-layer conformer encoder as an example.}\n    \\label{fig:linear_alg}\n\\end{figure}\n\nPrevious parameter-efficient fine-tuning methods update the sparse parameters in the foundation model and use the output of the highest encoder layer only as the input to the RNN-T decoder, while the outputs of the intermediate layers are dropped after the forward pass. The proposed feature fusion method treat the foundation model as a frozen feature extractor and fuse the multi-level features from different layers linearly or hierarchically. Because there is no need to perform backward pass in the foundation model and only a few parameters are added on top of the outputs of the intermediate layers, the proposed feature fusion method is parameter-efficient and computation-efficient.  \n\n\\subsection{Performance of Single Layer Features}\n\\label{sec:linear_1}\nTo study the performance of the features from different layers of the foundation model, we extract outputs from layers \\\\\n$\\{1, 3, 5, 10, 12, 14, 19, 21, 23\\}$ respectively and update the $124$M 6-layer LSTM decoder only on the Voice Search data. Figure \\ref{fig:linear_wer} shows the WER of the corresponding layers and results present that models using features from middle layers perform better on the speech recognition task than features from bottom or top layers. This observation is consistent with ~\\cite{pasad2021layer} that middle layers encode more contextual and high-level information which is more helpful for the speech recognition task than bottom or top layers. \n\n\\begin{figure*}[t]\n     \\centering\n     \\begin{subfigure}[b]{0.48\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{HFFb_v2.png}\n     \\end{subfigure}\n     \\begin{subfigure}[b]{0.48\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{HFFub_v2.png}\n     \\end{subfigure}\n        \\caption{Balanced and unbalanced hierarchical feature fusion methods of the foundation model, using $6$-layer conformer encoder as an example. FP denotes a $1$-layer fully-connected network.}\n    \\label{fig:hff_alg}\n\\end{figure*}\n\n\n\n\\subsection{Linear Feature Fusion From Multiple Layers}\nFrom Section \\ref{sec:linear_1}, we know that features from different layers show different performance on the downstream speech recognition task. To investigate whether these features are complementary, we propose a linear feature fusion method and combine features from different layers linearly. As in Figure \\ref{fig:linear_alg}, we firstly concatenate the features from multiple layers and project the concatenated feature to the required dimension using a fully-connected neural network. The decoder receives the output of the projector as input. All the conformer layers in the encoder are fixed while we update feature projector and decoder only using the RNN-T loss. \n\n\\begin{table}[h]\n    \\centering\n    \\caption{Fusing features from multiple layers of the foundation model. Feature projector is a  $1$-layer fully-connected network for all combinations. }\n    \\label{tab:linear_multiple}\n    \\begin{tabular}{ccc}\n        \\toprule\n        \\textbf{Layer index} & \\textbf{\\# Parameters} & \\textbf{VS WER} \\\\\n        & \\textbf{In Feature Projector} & {($\\%$)} \\\\\n        \\midrule\n         $11$ & $0.6$ M  & $11.2$ \\\\\n        \\midrule\n         $23$ & $0.6$ M  & $91.9$ \\\\\n        \\midrule\n         $11, 23$ & $1.3$ M  & $10.8$ \\\\\n        \\midrule\n         $5, 11, 17, 23$ & $2.6$ M  & $9.3$ \\\\\n        \\midrule\n         $2, 5, 8, 11, 14$ & $5.2$ M  & $8.1$ \\\\\n         $ 17, 20, 23$ &  & \\\\\n        \\midrule\n         $1, 3, 5, 7, 9, 11, 13$ & $7.9$ M  & $\\textbf{8.0}$ \\\\\n         $ 15, 17, 19, 21, 23$ & & \\\\\n        \\bottomrule\n    \\end{tabular}\n\\end{table}\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{linear_weight_layers.png}\n    \\caption{$\\ell_2$ norm of the learned weight of each layer when fusing features from $12$ layers.}\n    \\label{fig:linear_weight}\n\\end{figure}\n\nWe compare the  Voice Search (VS) WER when fusing features from $\\{1, 2, 4, 8, 12\\}$ layers. Results in Table \\ref{tab:linear_multiple} demonstrate the benefit of feature fusion from multiple layers. When fusing features from $12$ layers, we obtain the best VS WER $8.0\\%$ with additional $7.9$M parameters in the feature projector. Figure \\ref{fig:linear_weight} shows the norm of the learned weights for each layer when fusing $12$ features. The figure presents a higher weight for the middle layers and a lower weight for bottom or top layers. The results demonstrate that features from middle layers contribute more to the speech recognition task and adding features from bottom or top layers is also helpful.  \n\n\n\n\n\\subsection{Increasing Depth of The Feature Projector}\nWe also explore to learn non-linear feature fusion by increasing the depth of the feature projector in Figure \\ref{fig:linear_alg}. In Table \\ref{tab:linear_depth}, we increase the depth of the fully-connected network from $1$ to $4$ layers with ReLU  activation while extracting features from the same $12$ layers used in the previous experiments, which was found to give the best results. Experimental results show that the model gets a better WER with a deeper feature projector and the VS WER becomes saturated at about $7.4\\%$ after adding up to $3$ fully-connected layers. \n\n\\begin{table}[h]\n    \\centering\n    \\caption{Increasing depth of the feature projector. Fusing features from $12$ layers as it gives the best results. }\n    \\label{tab:linear_depth}\n    \\begin{tabular}{ccc}\n        \\toprule\n        \\textbf{\\# Layers} & \\textbf{\\# Parameters} & \\textbf{VS WER} \\\\\n        & \\textbf{In Feature Projector} & {($\\%$)} \\\\\n        \\midrule\n         $1$ & $7.9$ M  & $8.0$ \\\\\n        \\midrule\n         $2$ & $8.3$ M  & $7.5$ \\\\\n        \\midrule\n         $3$ & $8.7$ M  & $\\textbf{7.4}$ \\\\\n        \\midrule\n         $4$ & $9.1$ M  & $\\textbf{7.4}$ \\\\\n        \\bottomrule\n    \\end{tabular}\n\\end{table}\n\\section{Hierarchical Feature Fusion of the Foundation Model}\n\\label{sec:hff}\nKnowing that features from different layers encode different levels of information, we also explore to fuse features in a hierarchical way rather than linearly. In this section, we propose a hierarchical feature fusion method and compare it with other parameter-efficient fine-tuning algorithms.\n\n\n\\begin{table*}[t]\n    \\centering\n    \\caption{Comparison with baselines and prameter-efficient methods. $\\downarrow$ denotes the smaller the better. Second column shows the number of trainable parameters in the encoder only for the corresponding compared method and the whole $124$M LSTM decoder are trainable as well.}\n    \\label{tab:comparison}\n    \\begin{tabular}{ccccc}\n        \\toprule\n        \\textbf{Methods} & \\textbf{\\# Trainable } & \\textbf{Computational } & \\textbf{Training Speed } & \\textbf{VS WER } \\\\\n        & \\textbf{Encoder Params $\\downarrow$} & \\textbf{Memory Cost $\\downarrow$}& \\textbf{Examples\/Sec $\\uparrow$} & \\textbf{ ($\\%) \\downarrow$} \\\\\n        \\midrule\n         Fine-tune  all &  $606.6$ M  & $13567$ MB & $1270$ & $5.5$  \\\\\n        \\midrule\n         Fine-tune the highest encoder layer (FTHS) &  $25.4$ M  & $7563$ MB & $3616$ & $15.8$   \\\\\n        \\midrule\n         BitFit & $0.1$ M  & $12443$ MB & $2824$ &  $6.5$ \\\\\n        \\midrule\n         Adapter(d=$128$) at all layers & $6.4$ M & $12411$ MB & $2810$ & $6.4$  \\\\\n        \\midrule\n         Adapter(d=$256$) at all layers & $13.3$ M & $12455$ MB & $2802$ & $6.1$  \\\\\n        \\midrule\n         Adapter(d=$512$) at all layers & $25.9$ M & $12486$ MB & $2788$ & $6.1$  \\\\\n        \\midrule\n         Adapter(d=$128$) at layers &&&& \\\\\n         $\\{13, 15, 17, 19, 21, 23\\}$ & $2.3$ M  & $9340$ MB & $3251$ & $7.9$ \\\\\n        \\midrule\n        \\midrule\n        Linear Feature Fusion & $8.7$ M & $7573$ MB & $3610$ & $7.4$ \\\\\n        \\midrule\n        HFF-b & $12.3$ M  & $7648$ MB &  $3655$ & $7.0$ \\\\\n        \\midrule\n         HFF-b + Adapter(d=$128$) at layers & & & \\\\\n          $\\{13, 15, 17, 19, 21, 23\\}$   & $13.9$ M & $\\textbf{9653}$ MB & $\\textbf{3213}$ & $\\textbf{6.0}$ \\\\\n        \\midrule\n         HFF-b + Adapter(d=$128$) at all layers  & $18.6$ M & $12378$ MB & $\\textbf{2750}$ & $\\textbf{5.5}$ \\\\\n        \\bottomrule\n    \\end{tabular}\n\\end{table*}\n\n\n\\subsection{Hierarchical Feature Fusion From Multiple Layers}\n\\label{sec:hff_methods}\n\n\nAs in Figure \\ref{fig:hff_alg}, we compare two hierarchical feature fusion methods (balanced and unbalanced) for the speech foundation model. For the balanced feature fusion method (HFF-b), we project and concatenate the neighboring pair-wise features, treating all layers equally. For the unbalanced feature fusion method (HFF-ub), on the other hand, we project and concatenate the neighboring features from bottom to the middle and from top to the middle. The intuition is that the middle layers encode high-level information while the bottom or top layers encode low-level information, such that more encoding is required for the features from these layers.\n\n\\begin{table}[h]\n    \\centering\n    \\caption{Comparison between balanced and unbalanced hierarchical feature fusion methods. Fusing features from $12$ layers.\n    }\n    \\label{tab:hff_comp}\n    \\begin{tabular}{ccc}\n        \\toprule\n        \\textbf{Methods} & \\textbf{\\# Parameters} & \\textbf{VS WER} \\\\\n        & \\textbf{In Feature Projector} &  \\\\\n        \\midrule\n         HFF-b &  $12.3$ M  & $\\textbf{7.0}$ \\\\\n        \\midrule\n         HFF-ub & $12.3$  M  & $7.2$ \\\\\n        \\bottomrule\n    \\end{tabular}\n\\end{table}\nWe use a $1$-layer fully-connected network as FP in Figure \\ref{fig:hff_alg} and the projector in the ``Concat $\\&$ Project\" is a $3$-layer fully-connected network. The FP projects a $1024$-d feature to $512$-d, such that the feature dimension remains unchanged after concatenation.  Table \\ref{tab:hff_comp} shows that both methods achieve better VS WER than linear feature fusion, and HFF-b performs better than the HFF-ub on the speech recognition task with the same amount of parameters in the feature projector. Therefore,  we use balanced  hierarchical feature fusion (HFF-b) in the following experiments.\n\n\n\n\n\\subsection{Comparison with Parameter-Efficient Fine-Tuning Methods}\n\\label{sec:hff-comp}\n\nTo validate the proposed hierarchical feature fusion method, we compare it to several related algorithms. Specifically, we compare with two representative and strong\nparameter-efficient methods: BitFit~\\cite{zaken2021bitfit} and Adapter~\\cite{houlsby2019parameter}. Each adapter module is inserted after each conformer encoder layer and is a randomly initialized $2$-layer feed-forward network with the bottleneck dimension $d$ from $\\{128, 256, 512\\}$.  We also fine-tune the highest conformer encoder layer (FTHST) as a baseline, which is computation-efficient because no backpropagation is required for the lower encoder layers. The parameter-efficient methods are applied to fine-tune the $600$M conformer encoder only, and the whole randomly initialized $124$M LSTM decoder is also updated simultaneously. Because the LSTM decoders are the same for all compared methods, we only compare the number of trainable encoder parameters in the experiments regarding parameter efficiency. Although the best VS WER can be achieved if we fine-tune all parameters of the model, it costs too much computational memory $13567$MB and the training speed is very slow at $1270$ examples\/sec. Results in Table \\ref{tab:comparison} show that Adapter's performance is better than BitFit or FTHS, but gets stuck at $6.1$ VS WER even if increasing the bottleneck dimension from $128$ to $512$. However, the Adapter(d=128) at all layers's training speed is $22\\%$ slower and computational memory cost is $64\\%$ higher than FTHS. With a very similar computational memory cost and training speed to FTHS, HFF-b can improve VS WER from $15.8\\%$ to $7.0\\%$. Comparing with Adapter(d=128) at layers $\\{13, 15, 17, 19, 21, 23\\}$, HFF-b achieves better VS WER with $12\\%$ faster training speed and $18\\%$ lower computation memory cost.  Combining the HFF-b with Adapter(d=128) at layers $\\{13, 15, 17, 19, 21, 23\\}$, we can achieve better VS WER $6.0\\%$ than all compared parameter-efficient methods with fewer number of trainable parameters, less computational memory cost and faster training speed. If combining the proposed HFF-b with Adapter($d=128$) at all layers, we can achieve the same WER as fine-tuning all parameters of the RNN-T model  with $97\\%$ fewer trainable encoder parameters and $53\\%$ faster training speed. \n\\section{Conclusion}\n\\label{sec:conclusion}\nIn this paper, we analyze the behavior of features from different layers of the foundation model for speech recognition task and  propose a hierarchical feature fusion method for resource-efficient transfer learning from the speech foundation model. Extensive results demonstrate that it achieves promising performance on the speech recognition task with fewer trainable encoder parameters, less computational cost and faster training speed. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAfter the discovery of Fe-based superconductors (FBS)\\,\\cite{Kamihara}, many efforts have been devoted to searching for new FBS, resulting in the discovery of various new superconductors such as FeSe\\cite{Hsu}, (Ba,K)Fe$_2$As$_2$\\cite{Rotter} and LiFeAs\\cite{Wang}.\nTo date SmFeAs(O,F) shows the highest $T_{\\rm c}$ of 58\\,K\\,\\cite{Fujioka} except for mono-layer FeSe\\,\\cite{Ge}. The common structural feature of FBS is the FeAs and Fe$Ch$ ($Ch$: chalcogen) tetrahedron, which is believed to play an important role for the superconductivity.\n$Ln$FeAs(O,F) ($Ln$: lanthanoide), $Ae$Fe$_2$As$_2$ ($Ae$: alkali earth elements) and  $A$FeAs ($A$: Li and Na) are formed with alternating layers of $Ln$O\/FeAs, $Ae$\/FeAs and $A$\/FeAs, and Fe$Ch$ is composed of a stack of Fe$Ch$ tetrahedra layer. Hence, FBS are expected to be anisotropic in their electronic properties, which may be closely connected with the superconductivity of these compounds. Therefore, investigating the electronic anisotropy may gain an important clue for the mechanism of high-$T_{\\rm c}$ superconductivity.\n\n\nTo date, large single crystals of some of the FBS systems have been grown and the temperature dependence of in-plane and out-of-plane resistivity [$\\rho_{ab}(T)$ and $\\rho_{c}(T)$] and the resultant resistivity anisotropy $\\gamma_\\rho(T)$ have been measured. A single crystal of FeSe$_{1-x}$Te$_x$ ($x$=0.6) with $T_{\\rm c}$=14.2\\,K grown by a Bridgeman method showed a metallic behaviour for $\\rho_{ab}$ below 140\\,K\\,\\cite{Noji}, whilst it was almost constant above 140\\,K. On the other hand, $\\rho_{c}$ increased gradually with decreasing temperature down to around 100\\,K and then decreased almost linearly with decreasing $T$. The resistivity anisotropy $\\gamma_ \\rho$ was 44 and 70 at 290\\,K and $T_{\\rm c}$, respectively.\n\n\nSong $et$ $al$. reported on the out-of-plane and in-plane resistivity for a LiFeAs single crystal with $T_{\\rm c}$=19.7\\,K grown by a Bridgeman method\\,\\cite{Song}. Below 200\\,K, $\\rho_{ab}(T)$ and $\\rho_{c}(T)$ decreased with lowering $T$, whilst the slopes of both resistivity curves decreased close to $T_{\\rm c}$. $\\gamma_ \\rho$ increased from 1.3 at 300\\,K to 3.3 at $T_{\\rm c}$.\n\n\nFor the BaFe$_2$As$_2$ systems, extensive electrical resistivity measurements on electron, hole and isovalent \ndoped single crystals with various doping levels have been reported\\,\\cite{Tantar-1, Tantar-2, Tantar-3}. The $ab$-plane resistivity for all optimally-doped BaFe$_2$As$_2$ single crystals decreased linearly from 300\\,K down to $T_{\\rm c}$. Similarly, the resistivity along the $c$-axis also showed a metallic behaviour at low temperature. However, the temperature range of the $T$-linear dependence was different among the BaFe$_2$As$_2$ systems: for P-doped BaFe$_2$As$_2$, $\\rho_{c}(T)$ was close to linear in the whole temperature range (i.e. from 300\\,K to down to $T_{\\rm c}$). On the other hand, $\\rho_{c}(T)$ for Co- and K-doped BaFe$_2$As$_2$ showed a broad maximum around 100\\,K and 220\\,K, respectively.   \n\n \nOn the other hand, the size of available $Ln$FeAsO single crystals is still limited. Only three papers  - to the best of our knowledge - regarding the temperature-dependence of $\\gamma_ \\rho$, PrFeAsO$_{0.7}$\\,\\cite{Kashiwaya}, SmFeAs(O,F)\\,\\cite{Moll} and SmFeAsO$_{0.9}$H$_{0.1}$\\,\\cite{Iimura}, have been published to date due to the difficulty in the crystal growth of this system. Additionally, no studies of the doping dependence of $\\gamma_ \\rho$ have been reported.\nBecause the dimensions of the obtained single crystals were typically $\\sim200\\times200\\times10$\\,$\\mu{\\rm m}^3$, a tiny bar was formed for resistivity measurements in both crystallographic main directions by a focused ion beam technique. For all three compounds, the $ab$-plane resistivity decreased almost linearly down to $T_{\\rm c}$, whereas along the $c$-axis the resistivity increased continuously to $T_{\\rm c}$, which is different from the results reported for other FBS. The respective resistivity anisotropy $\\gamma_ \\rho$ at 50\\,K for PrFeAsO$_{0.7}$, SmFeAs(O,F) and SmFeAsO$_{0.9}$H$_{0.1}$ were 120, 8.4 and 7.8.  \n\n\nAs stated above, the anisotropic behaviour of the transport properties of FBS has been investigated using single crystals. \nHowever, the number of studies on $Ln$FeAsO systems are limited and the obtained results are different from each other. Additionally, no studies of the doping dependence of $\\gamma_ \\rho$ have been reported.\nHere, we report on the transport properties of NdFeAs(O,F) thin films with different F-doping levels grown on vicinal substrates for which the [001] direction is 5$^\\circ$ or 10$^\\circ$ tilted toward the [100] direction. Using these off-axis grown thin films, we evaluate the resistivity anisotropy, adopting the method that has been employed to study the anisotropy of Bi$_2$Sr$_2$CaCu$_2$O$_8$\\,\\cite{Zahner}, YBa$_2$Cu$_3$O$_7$ (YBCO)\\,\\cite{Haage, Czerwinka, Emergo}, MgB$_2$\\,\\cite{Polyanskii} and Fe(Se,Te)\\,\\cite{Bryja}. \n\n\\section{Experiment}\nParent NdFeAsO thin films having a thickness of 28--90\\,nm were grown on vicinal cut MgO(001) and CaF$_2$(001) single crystalline substrates at 800$^\\circ$C by molecular beam epitaxy (MBE). The nominal vicinal angle $\\theta_{\\rm vic}$ was 5$^\\circ$ or 10$^\\circ$ measured from the substrate normal toward the [001] directions, as shown in fig.\\,\\ref{fig:figure1}(a). The film growth was monitored by reflection high-energy electron diffraction (RHEED). To obtain NdFeAs(O,F) films with different F content, a 20\\,nm-thick NdOF over-layer was deposited on top of the NdFeAsO layer at various temperatures in the range of $550^\\circ{\\rm C}\\leq T_{\\rm dep} \\leq 800^\\circ$C\\,\\cite{Kawaguchi-1, Iida-1}, where $T_{\\rm dep}$ is the deposition temperature.  Since it is difficult to determine the fluorine content precisely, both the $c$-axis lattice parameter and $T_{\\rm c}$ have been used as the indicators of the F-content in NdFeAs(O,F). Additionally, the carrier concentration of some of the films was measured by Hall effect. To rule out the possibility that the NdOF over-layer affected the transport properties, NdOF was removed for some of the NdFeAs(O,F) films by Ar-ion beam etching. We confirmed that the presence or absence of the NdOF over-layer gives no difference in the transport properties of NdFeAs(O,F) (Supplementary figure\\,\\ref{fig:figureS1}).\n\n\nPhase purity and the out-of-plane texture were measured by x-ray diffraction (XRD) in Bragg-Brentano geometry using Cu-K$\\alpha$ radiation. The growth angle $\\alpha$ (i.e. offset angle) was determined as the tilt angle where the intensity maxima was observed for the 003 reflection. In-plane orientation of NdFeAs(O,F) was investigated by $\\phi$-scans of the 200 peak. \n\n\n\\begin{figure}[ht]\n\t\\centering\n\t\t\\includegraphics[width=\\columnwidth]{Figure1.pdf}\n\t\t\\caption{(a) Schematic illustration of a vicinal substrate for which the [001] direction is away from the substrate normal by $\\theta_{\\rm vic}$. (b) An example of a top-view optical micrograph for the micro-bridges. A bridge running transverse to the vicinal steps is defined as a T-bridge, whereas a bridge parallel to the steps is defined as a L-bridge.} \n\\label{fig:figure1}\n\\end{figure}\n\n\n\\begin{table*}[bt]\n\\centering\n\\caption{\\label{tab:table1}Sample name, $T_{\\rm dep}$ of NdOF, the ratio of the thickness of NdOF ($d_{\\rm NdOF}$) and NdFeAsO ($d_{\\rm NdFeAsO}$), the $c$-axis lattice parameter, the growth angle (offset angle) and the onset $T_{\\rm c}$ ($T_{\\rm c}^{\\rm onset}$) of the samples used in this study. The sample nomenclature is based on the combination of the doping level, the value of onset $T_{\\rm c}$ and the substrate. ``OP\" represents the optimum doping level at which the NdFeAs(O,F) films show $T_{\\rm c}$ higher than 46\\,K. The NdFeAs(O,F) films having $T_{\\rm c}$ below 44\\,K are defined as ``UD\". ``PC\" represents the parent compound, NdFeAsO.}\n\n\\begin{ruledtabular}\n\\begin{tabular}{cccccc}\nSample name & $T_{\\rm dep}$ ($^\\circ$C) & $d_{\\rm NdOF}$\/$d_{\\rm NdFeAsO}$ & $c$-axis\\,(nm) & Growth angle $\\alpha(^\\circ)$ & $T_{\\rm c}^{\\rm onset}$(K) \\\\ \\hline\nPC\/MgO          &-     &0\/45    &0.8590   &12.25&-\\\\\nPC\/CaF$_2$   &-     &0\/28    &0.8669   &5.17  &-\\\\\nUD0\/MgO         &550&20\/30  &0.8591   &6.28   &-\\\\\nUD0\/CaF$_2$  &700&20\/90  &0.8681   &5.23   &-\\\\\nUD36\/MgO       &650&20\/90  &0.8576   &6.91&36\\\\\nUD42\/MgO       &650&20\/30  &0.8586   &5.80&42\\\\\nUD43\/MgO       &750&20\/90  &0.8569   &6.85&43\\\\\nUD44\/CaF$_2$&800&20\/90  &0.8663   &5.18&44\\\\\nOP45\/MgO       &800&20\/30  &0.8533   &5.23&45\\\\\nOP46\/MgO       &800&20\/30  &0.8535   &11.68&46\\\\\nOP49\/MgO       &800&20\/90  &0.8540    &6.33&49\\\\\nOP56\/CaF$_2$&800&20\/90  &0.8619    &5.06&56\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table*}\n\nAfter structural characterisations, the films were photolithographically patterned and etched by Ar-ion milling to form micro-bridges for transport measurements. \nAs shown in fig.\\,\\ref{fig:figure1}(b), the bridges were designed to measure the resistivity in the longitudinal direction (abbreviated as L-bridge and the corresponding resistivity $\\rho_{\\rm L}$) and in the transversal direction (abbreviated as T-bridge and the corresponding resistivity $\\rho_{\\rm T}$). In the longitudinal direction, the bias current flows within the $ab$-plane only (i.e. $\\rho_{\\rm L}$=$\\rho_{ab}$), whereas in the transversal direction it flows within the $ab$-plane as well as along the $c$-axis.  \nTherefore, the $c$-axis resistivity can be calculated by the following equation\\,\\cite{Zahner},\n\n\\begin{equation}\n\\rho_{c}=(\\rho_{\\rm T}-\\rho_{\\rm L}{\\rm cos}^2\\alpha)\/{\\rm sin}^2\\alpha.\n\\end{equation}\n\n\n\\noindent\nA precision of 0.005$^\\circ$\/2$\\theta$ is guaranteed for our XRD device, which yields a relative uncertainty of $\\left| \\frac{\\delta \\rho_c}{\\rho_c} \\right|$$\\sim$0.05\\% estimated using the typical values of $\\rho_{\\rm T}$ and $\\rho_{\\rm L}$ of our films. \nMisalignment may also occur when the sample was mounted on the sample holder. We checked the XRD data of our recent films grown on ordinary single crystalline substrates and found that the misalignment angle was at most 0.020$^\\circ$\/2$\\theta$. This yields a relative uncertainty of $\\left| \\frac{\\delta \\rho_c}{\\rho_c} \\right|$$\\sim$0.39\\%. Together, the maximum uncertainty due to the inaccuracy of $\\alpha$ is $\\left| \\frac{\\delta \\rho_c}{\\rho_c} \\right|$$\\sim$0.50\\% in our measurements.\n The bridges had dimensions 50\\,$\\mu{\\rm m}$-wide and 0.5\\,mm-long. \nFor measuring the angular-dependence of $\\rho_{ab}$, a magnetic field, $H$, was applied in the maximum Lorentz force configuration ($H$ perpendicular to the bias current) at an angle $\\theta$ measured from the $c$-axis. For some of the NdFeAs(O,F) films, Hall resistance was measured in the field range of $\\mu_0H=\\pm9$\\,T to evaluate the carrier concentration. The samples studied here are summarised in table\\,\\ref{tab:table1}. \n\n\n\\section{Results and discussion} \n\\begin{figure}[ht]\n\t\\centering\n\t\t\\includegraphics[width=\\columnwidth]{Figure2.pdf}\n\t\t\\caption{(a) A representative RHEED image of NdFeAsO during the film growth. (b) The $\\theta\/2\\theta$-scan (Cu-K$\\alpha$) of the NdFeAs(O,F) (Nd1111) thin film grown on a vicinal MgO substrate measured with an offset angle of 6.33$^\\circ$. (c) The 005 rocking curve and (d) the 200 $\\phi$-scan of the film shown in fig.\\,(b)} \n\\label{fig:figure2}\n\\end{figure}\n\n\nThe RHEED image of NdFeAsO during the film growth of OP49\/MgO showed long streaky patterns tilted by $\\sim$5$^\\circ$, indicative of an epitaxial growth with a smooth surface [fig.\\,\\ref{fig:figure2}(a)]. This RHEED pattern maintained until the termination of the growth of NdFeAsO. We confirmed that all RHEED images of the NdFeAsO layers showed streaky patterns irrespective of the kinds of substrates as well as the vicinal angles. Figure\\,\\ref{fig:figure2}(b) shows the XRD pattern of the corresponding film presented in fig.\\,\\ref{fig:figure2}(a) measured with an offset angle of 6.33$^\\circ$, at which the diffraction intensity of the 003 reflection was maximum (the nominal vicinal angle=5$^\\circ$). The deposition temperature of NdOF was 800$^\\circ$C. Due to the offset angle, the data below 13$^\\circ$ were absent. Pronounced 00$l$ diffraction peaks from NdFeAs(O,F) and NdOF were observed, indicating $c$-axis orientation for both compounds. The $c$-axis lattice parameter of NdFeAs(O,F) calculated using the Nelson-Riley function\\,\\cite{Nelson} was 0.8540\\,nm, which is smaller than the parent NdFeAsO film on MgO substrate ($c$=0.8590\\,nm, see table\\,\\ref{tab:table1}). The shortening of the $c$-axis length indicates that F was doped into NdFeAsO, resulting in the formation of the NdFeAs(O,F) phase.\nIndeed, the Hall measurements revealed an increase in the carrier density $n$ for OP49\/MgO compared from that of parent NdFeAsO: $n$ at 50\\,K for OP49\/MgO was 1.98$\\times$10$^{21}$\\,cm$^{-3}$ when estimated using a single carrier model, whereas the corresponding value for NdFeAsO was 0.02$\\times$10$^{21}$\\,cm$^{-3}$, although the precise evaluation of $n$ is difficult due to the multi-band nature of FBS. The rocking curve of the 005 reflection for OP49\/MgO showed a full width at half maximum $\\Delta \\omega$ of 0.89$^\\circ$, indicative of a good out-of-plane texture [fig.\\,\\ref{fig:figure2}(c)]. The 200 $\\phi$-scan of the NdFeAs(O,F) film exhibited two peaks separated by 180$^\\circ$ [fig.\\,\\ref{fig:figure2}(d)], which differs from an epitaxial NdFeAs(O,F) film on an ordinary MgO substrate. This is because the rotational axis for $\\phi$-scan is tilted away from the crystallographic $c$-axis. These results proved that the film was epitaxially grown with the $c$-axis tilted by 6.33$^\\circ$. We confirmed that all films investigated in this study were epitaxially grown. \n\n\n\\begin{figure}[ht]\n\t\\centering\n\t\t\\includegraphics[width=\\columnwidth]{Figure3.pdf}\n\t\t\\caption{The resistivity curves for all samples tabulated in table\\,\\ref{tab:table1} along the transversal and longitudinal directions [$\\rho_{\\rm T}(T)$ and $\\rho_{\\rm L}(T)$].} \n\\label{fig:figure3}\n\\end{figure}\n\nFigure\\,\\ref{fig:figure3} summarises the temperature dependence of the resistivity in the transversal and longitudinal directions [$\\rho_{\\rm T}(T)$ and $\\rho_{\\rm L}(T)$] for all samples shown in table\\,\\ref{tab:table1}.\nThe measured resistivity in the transversal direction $\\rho_{\\rm T}(T)$ was always higher than in the longitudinal direction $\\rho_{\\rm L}(T)$ due to the anisotropic electronic structure of NdFeAs(O,F). For the superconducting thin films, both transversal and longitudinal bridges had almost the same $T_{\\rm c}$, proving that all films were homogeneous. Another distinct feature is that $\\rho_{\\rm T}(T)$ increased with increasing the vicinal angle due to the increase of the $c$-axis component. This can be clearly seen in the normalised resistivity traces for OP45\/MgO and OP46\/MgO (Supplementary figure\\,\\ref{fig:figureS2}). As can be seen, the difference between $\\rho_{\\rm T}$ and $\\rho_{\\rm L}$ for OP45\/MgO is larger than that for OP46\/MgO because of the larger tilt angle.\n\n\nThe $c$-axis resistivity $\\rho_{c}$ can be calculated by using eq. (1) and the growth angle $\\alpha$ [fig.\\,\\ref{fig:figure4}(a)]. For the parent NdFeAsO and under-doped NdFeAs(O,F) films, a kink in the temperature dependence of $\\rho_{c}$ due to the structural transition was observed at around 150\\,K. Below the structural transition temperature ($T_{\\rm str}$), $\\rho_{c}$ decreased with lowering $T$. Here, a distinct feature was observed for the films on CaF$_2$ substrates: For both PC\/CaF$_2$ and UD0\/CaF$_2$ $\\rho_{c}$ started to increase at around 30\\,K with decreasing temperature. Such behaviour was not observed for the corresponding films grown on MgO substrates. For all superconducting UD samples $\\rho_{c}(T)$ decreased below $T_{\\rm str}$, and a gradual decrease of $d\\rho_{c}\/dT$ close to $T_{\\rm c}$ was recognised. Similar behaviour was observed in other FBS single crystals like Ba$_{0.81}$K$_{0.19}$Fe$_2$As$_2$, BaFe$_2$(As$_{0.77}$P$_{0.23}$)$_2$ (i.e. under-doped regime)\\,\\cite{Tantar-2}, and LiFeAs\\,\\cite{Song}. For all OP samples, on the other hand, the temperature-dependent $\\rho_{c}(T)$ was metallic, which differs from the reports on PrFeAsO$_{0.7}$\\,\\cite{Kashiwaya}, SmFeAs(O,F)\\,\\cite{Moll} and SmFeAsO$_{0.9}$H$_{0.1}$\\,\\cite{Iimura} single crystals. To identify the reason of this difference, it is desired to measure $\\rho_{c}(T)$ using NdFeAs(O,F) single crystals. However, the size of available single crystals is limited. Further investigation is necessary.\n\n\n\\begin{figure}[b]\n\t\\centering\n\t\t\\includegraphics[width=\\columnwidth]{Figure4.pdf}\n\t\t\\caption{(a) The $c$-axis resistivity curves for all samples tabulated in table\\,\\ref{tab:table1}. (b) The temperature-dependency of resistivity anisotropy\n\t\t $\\gamma_{\\rho}(T)=\\rho_{c}\/\\rho_{ab}$ for the films presented in (a).} \n\\label{fig:figure4}\n\\end{figure}\n\n\nIn fig.\\,\\ref{fig:figure4}(b) the temperature-dependent resistivity anisotropy $\\gamma_{\\rho}(T)=\\rho_{c}\/\\rho_{ab}$ for all samples is shown. \nAgain distinct features were observed for PC\/CaF$_2$ and UD0\/CaF$_2$: 1) $\\gamma_{\\rho}$ for PC\/CaF$_2$ was almost constant irrespective of $T$. On the other hand for PC\/MgO and UD0\/MgO $\\gamma_{\\rho}$ increased with decreasing $T$ down to $T_{\\rm str}$ and decreased thereafter. A similar beahviour was also observed for UD36\/MgO. 2) For UD0\/CaF$_2$ $\\gamma_{\\rho}$ increased with decreasing $T$ down to $T_{\\rm str}$, which is similar to UD0\/MgO, but increased further rather rapidly below $T_{\\rm str}$, which is different from UD0\/MgO.  \n\n\nFor the under-doped superconducting samples except UD36\/MgO, $\\gamma_{\\rho}$ increased monotonously with decreasing $T$ down to $T_{\\rm c}$ irrespective of the substrate. The increase was especially strong for the OP series samples, and showed an exponential-like $T$ dependence. For OP49\/MgO and OP56\/CaF$_2$, $\\gamma_{\\rho}$ were more than 220 near $T_{\\rm c}$, which were quite large values compared to single crystals. Note that $\\rho_{c}(T)$ was metallic for both OP49\/MgO and OP56\/CaF$_2$, and the increase of $\\gamma_{\\rho}(T)$ with lowering $T$ is because the rate of decrease of $\\rho_{ab}(T)$ was much faster than $\\rho_{c}(T)$.  \n\n\n\\begin{figure}[ht]\n\t\\centering\n\t\t\\includegraphics[width=\\columnwidth]{Figure5.pdf}\n\t\t\\caption{(a) Angular dependence of $\\rho_{ab}$ for the NdFeAs(O,F) film OP49\/MgO at 39\\,K in various magnetic fields. (b) The scaling behaviour of $\\rho_{ab}(\\theta)$ as a function of $H_{\\rm eff}$. (c) The mass anisotropy $\\gamma_m$ derived from the anisotropic Ginzburg-Landau approach as a function of temperature for all superconducting films. (d) The plots of the square root of $\\gamma_{\\rho}$($T$) shown in fig.\\,\\ref{fig:figure4}(b) and $\\gamma_m(T)$.} \n\\label{fig:figure5}\n\\end{figure}\n\nIn order to evaluate the mass anisotropy in the superconducting state, the angular dependence of resistivity $\\rho_{ab}(\\theta)$ was measured \nbelow $T_{\\rm c}$. Figure\\,\\ref{fig:figure5}(a) shows the results for OP49\/MgO measured at 39\\,K in various magnetic fields. On the assumption that no correlated defects are present, the mass anisotropy governs the resistivity anisotropy. In this case, $\\rho_{ab}(\\theta)$ can be scaled with $H_{\\rm eff}$ $[H_{\\rm eff}=H\\epsilon(\\theta)$, $\\epsilon(\\theta)=\\sqrt{\\cos ^2\\theta+\\gamma_m^{-2}\\sin ^2\\theta}$], where $\\gamma_m$ is the mass anisotropy\\,\\cite{Blatter}. This approach was applied to a NdFeAsO$_{0.82}$F$_{0.18}$ single crystal ($T_{\\rm c}$=51.5\\,K) and its $\\rho_{ab}(\\theta)$ at given temperatures were scaled with an appropriate $\\gamma_m$, which increased with lowering $T$\\,\\cite{Jia-1}.\nNear $T_{\\rm c}$, $\\gamma_m$ was 5.2, which was consistent with the value evaluated from the ratio of the upper critical field $H_{\\rm c2}^{ab}\/H_{\\rm c2}^{c}$\\,\\cite{Jia-2}. \n\n\nThe scaling behaviour of $\\rho_{ab}(\\theta)$ for OP49\/MgO  as a function of $H_{\\rm eff}$ is shown in fig.\\,\\ref{fig:figure5}(b). It is clear that all data collapsed onto a single curve with $\\gamma_ m$=2.1. We also evaluated $\\gamma_m$ at different temperatures as well as for all superconducting films. As can be seen, the resulting $\\gamma_m$ was found to decrease with decreasing temperature [fig.\\,\\ref{fig:figure5}(c)]. This observation differs from the result obtained from the single crystal\\,\\cite{Jia-1}. It is interesting to note that $\\gamma_ m$ for the under-doped films are higher than those for the optimal-doped films. This may be due to the decrease in the interlayer coupling, which was similarly observed in cuprates\\,\\cite{Kishio}.\n\n\nBy assuming $\\rho_{ab}=m^*_{ab}\/ne^2\\tau$ and $\\rho_{c}=m^*_{c}\/ne^2\\tau$, where $n$ is the carrier density, $e$ the electric charge, \nand $\\tau$ the relaxation time of carriers, the mass anisotropy $\\gamma_m^2$ should be equal to $\\gamma_ \\rho$ at $T_{\\rm c}$. The mass anisotropy and square root of $\\gamma_{\\rho}$ as a function of temperature are shown in fig.\\,\\ref{fig:figure5}(d). As can be seen, the temperature dependence of the anisotropy showed a discontinuous change at $T_{\\rm c}$. The difference between $\\gamma_\\rho^{0.5}$ and $\\gamma_m$ is getting larger with increasing the F content. Similar discrepancy of the anisotropy at $T_{\\rm c}$ was observed in YBCO. The resistivity anisotropy obtained from a vicinal YBCO film at 100\\,K was around 60\\,\\cite{Emergo}, which is in good agreement with the value obtained from twin free single crystals\\,\\cite{Friedmann}. On the other hand, $\\gamma_m$ obtained from the anisotropic Ginzburg-Landau approach for clean YBCO films was around 5\\,\\cite{Civale}, which differs from the values derived from the resistivity anisotropy. Albeit the reason why the mass anisotropy is different in the normal and superconducting states for YBCO is unclear, such difference may be possible for FBS due to their multi-band nature. If the dominant electronic bands which govern the transport properties are different in the normal and superconducting states, the mass anisotropy could be different in the two states. This may partially explain our experimental results.\n\n\\section{Summary} \nNdFeAs(O,F) epitaxial thin films having different F contents were grown successfully on vicinal cut MgO and CaF$_2$ substrates by MBE. \nUsing these films the anisotropy of the electrical transport properties were investigated. The $c$-axis resistivity was always higher than $\\rho_{ab}$ irrespective of the F content, resulting from the anisotropic electronic structure. The temperature-dependent $\\rho_{c}(T)$ for superconducting films was metallic at low temperature, which differs from the other reports on the $Ln$FeAsO system. In the superconducting state, the mass anisotropy for the superconducting thin films were derived using the anisotropic Ginzburg-Landau approach. Near $T_{\\rm c}$, the resultant values are different to these obtained from the resistivity anisotropy, which may be due to the multi-band nature of NdFeAs(O,F).\n\n\\begin{acknowledgments}\nThis work was supported by the JSPS Grant-in-Aid for Scientific Research (B) Grant Number 16H04646 as well as JST CREST Grant Number JPMJCR18J4. \n\\end{acknowledgments}\n  \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA dynamical system is \\emph{finitely presented} if it can be\nrepresented as the factor of a shift of finite type by an equivalence\nrelation that is also a shift of finite type,\nsee~\\cite{fried,curnpap:symb}. \nIf we think of finite type as analogous\nto finite generation (of a group or of a normal subgroup), then the\nnotion of a finitely presented dynamical system becomes analogous to the notion of a finitely\npresented group. But the relation is deeper than just a superficial\nanalogy.\n\nCondition of being finitely presented for a dynamical system is very\nclosely related to dynamical hyperbolicity, see~\\cite{fried}. For\nexample, if $f:\\mathcal{J}\\longrightarrow\\mathcal{J}$ is a locally expanding self-covering of a\ncompact metric space, then $f$ is finitely presented. Dynamical\nhyperbolicity is very closely related to Gromov hyperbolicity for\ngroups, see~\\cite{gro:hyperb,curnpap:symb,nek:hyperbolic}, and finite\npresentation is an important property of hyperbolic groups.\n\nThe aim of this paper is to show a new connection between expanding\nmaps and finite presentations. We naturally associate with every\nfinite degree self-covering\n$f:\\mathcal{M}\\longrightarrow\\mathcal{M}$ of a path-connected space $\\mathcal{M}$ a group $\\mathcal{V}_f$ with the\nfollowing property (see Theorem~\\ref{th:finitepresentation} and Theorem~\\ref{th:classification}).\n\n\\begin{theorem}\n\\label{th:first}\nIf $f:\\mathcal{M}\\longrightarrow\\mathcal{M}$ is a locally expanding self-covering of a compact\npath connected metric space then $\\mathcal{V}_f$ is finitely presented.\n\nIf $f_i:\\mathcal{M}_i\\longrightarrow\\mathcal{M}_i$ are as above, then groups $\\mathcal{V}_{f_1}$ and\n$\\mathcal{V}_{f_2}$ are isomorphic if and only if $f_1$ and $f_2$ are\ntopologically conjugate, i.e., there exists a homeomorphism\n$\\phi:\\mathcal{M}_1\\longrightarrow\\mathcal{M}_2$ such that $f_1=\\phi^{-1}\\circ f_2\\circ\\phi$.\n\\end{theorem}\n\nWe also show that the commutator subgroup of $\\mathcal{V}_f$ is simple, and\ngive a dynamical interpretation of the abelianization $\\mathcal{V}_f\/\\mathcal{V}_f'$,\nsee Theorem~\\ref{th:commutatorsimple} and\nProposition~\\ref{pr:homology}.\n\nThe groups $\\mathcal{V}_f$ are defined in the following way. Let $f:\\mathcal{M}\\longrightarrow\\mathcal{M}$\nbe a finite degree covering map, and suppose that $\\mathcal{M}$ is path\nconnected. Choose $t\\in\\mathcal{M}$, and consider the tree $T_t$ of preimages of $t$\nunder the iterations of $f$. Its set of vertices is $\\bigsqcup_{n\\ge\n  0}f^{-n}(t)$, and a vertex $v\\in f^{-n}(t)$ is connected to the\nvertex $f(v)\\in f^{-(n-1)}(t)$. Let $\\partial T_t$ be its boundary,\nwhich can be defined as the\ninverse limit of the discrete sets $f^{-n}(t)$ with respect to the\nmaps $f:f^{-(n+1)}(t)\\longrightarrow f^{-n}(t)$.\n\nLet $\\gamma$ be a path in $\\mathcal{M}$ connecting a vertex $v\\in f^{-n}(t)$ to\na vertex $u\\in f^{-m}(t)$ of the tree $T_t$. Considering lifts of\n$\\gamma$ by the coverings $f^k:\\mathcal{M}\\longrightarrow\\mathcal{M}$, we get an isomorphism\n$S_\\gamma:T_v\\longrightarrow T_u$ between subtrees $T_v, T_u$ of $T_t$. Namely,\nif $\\gamma_z$ is a lift of $\\gamma$ starting at $z\\in f^{-k}(v)$, then\n$S_\\gamma(z)\\in f^{-k}(u)$ is the end of $\\gamma_z$.\nDenote by the same symbol\n$S_\\gamma$ the induced homeomorphism $\\partial T_v\\longrightarrow\\partial T_u$ of\nthe boundaries of the subtrees, seen as clopen subsets of $\\partial T_t$.\n\n\\begin{defi}\nThe group $\\mathcal{V}_f$ is the group of all homeomorphisms $\\partial\nT_t\\longrightarrow\\partial T_t$ locally equal to homeomorphisms of the form\n$S_\\gamma:\\partial T_v\\longrightarrow\\partial T_u$.\n\\end{defi}\n\nThe group $\\mathcal{V}_f$ is generated by the Higman-Thompson group $G_{\\deg f,\n1}$ (see~\\cite{hgthomp}) acting on $\\partial T_t$ and the \\emph{iterated monodromy group}\n$\\img{f}$ of $f:\\mathcal{M}\\longrightarrow\\mathcal{M}$. The iterated monodromy group can be defined\nas the subgroup of $\\mathcal{V}_f$ consisting of homeomorphism\n$S_\\gamma:\\partial T_v\\longrightarrow\\partial T_v$,\nwhere $\\gamma$ is a loop starting and ending at the basepoint $t$.\nIt is an\ninvariant of the topological conjugacy class of $f:\\mathcal{M}\\longrightarrow\\mathcal{M}$, and it\nbecomes a complete invariant (in the expanding case), if we consider\nit as a \\emph{self-similar group}. Self-similarity is an additional\nstructure on a group, and it can be defined using one of several\nequivalent approaches: virtual endomorphisms, wreath recursions,\nbisets, or structures of an automaton group. The fact that self-similar\niterated monodromy group is a complete invariant of an expanding\nself-covering is one of the main topics of~\\cite{nek:book}.\n\nFrom the point of view of group theory, $\\mathcal{V}_f$ seems to be a\n``cleaner'' object, since no additional structure is needed to make\nit a complete invariant of a dynamical system. Besides, it is finitely\npresented, unlike the iterated monodromy groups, which are typically\ninfinite presented. \nHowever, iterated monodromy groups have better functorial properties\nthan the groups $\\mathcal{V}_f$, see~\\cite{nek:filling}.\n\nA group $\\mathcal{V}_G$, analogous to $\\mathcal{V}_f$, can be defined for any self-similar\ngroup $G$, so that $\\mathcal{V}_f=\\mathcal{V}_{\\img{f}}$. Groups of this type were for\nthe first time studied by C.~R\\\"over~\\cite{roever,roever:comm}. In particular, he showed\nthat if $G$ is the Grigorchuk group~\\cite{grigorchuk:80_en}, then $\\mathcal{V}_G$ is finitely\npresented, simple, and is isomorphic to the abstract commensurator of\nthe Grigorchuk group. The case of a general self-similar group $G$ was\nstudied later in~\\cite{nek:bim}.\n\nSeveral natural questions arise in connection with\nTheorem~\\ref{th:first}. For example, is the isomorphism problem\nsolvable for the groups $\\mathcal{V}_f$? Equivalently, is the topological\nconjugacy problem for expanding maps algorithmically solvable? \nNote that expanding maps can\nbe given in different ways by a finite amount of information: using\nfinite presentations in the sense of~\\cite{fried}, using combinatorial\nmodels in the sense of~\\cite{ishiismillie,nek:models}, using iterated monodromy\ngroups, e.t.c..\n\nAnother natural question is whether the groups $\\mathcal{V}_f$, similarly to the\nHigman-Thompson groups (see~\\cite{brown:finiteness}), \nsatisfy the finiteness condition $F_\\infty$,\ni.e., if they have classifying spaces with finite $n$-dimensional\nskeleta for all $n$.\nIt would be also interesting to study homology of $\\mathcal{V}_f$ in relation with\nhomological properties of the dynamical system.\n\nThe structure of the paper is as follows. In \n``Definition of the groups $\\mathcal{V}_f$'' we give a review of terminology\nrelated to\nrooted trees, and define the groups $\\mathcal{V}_f$. In the next section\n``Symbolic coding'' we encode the vertices of the tree of preimages\n$T_t$ by finite words over an alphabet $X$, and show that $\\mathcal{V}_f$\ncontains a natural copy of the Higman-Thompson group, and that $\\mathcal{V}_f$\nis generated by the Higman-Thompson group and the iterated monodromy\ngroup. We also give a review of the basic notions of the theory of\nself-similar groups, and define the groups $\\mathcal{V}_G$ associated with\nself-similar groups, following~\\cite{nek:bim}.\n\nIn Section~4 we prove that the commutator subgroup $\\mathcal{V}_G'$ of $\\mathcal{V}_G$\nis simple for any self-similar group $G$. In particular, $\\mathcal{V}_f'$ is\nsimple for any map $f$.\nNote that the fact that every proper quotient of $\\mathcal{V}_G$ is abelian,\ni.e., that every non-trivial normal subgroup of $\\mathcal{V}_G$ contains\n$\\mathcal{V}_G'$ was already proved in~\\cite{nek:bim}, and we use this fact in\nour proof. Later, in the next section we give an interpretation of $\\mathcal{V}_f\/\\mathcal{V}_f'$ in\ntopological terms. Namely, we prove the following (see Proposition~\\ref{pr:homology}).\n\n\\begin{proposition}\nSuppose that $f:\\mathcal{M}_1\\longrightarrow\\mathcal{M}$ is expanding, $\\mathcal{M}$ is path-connected and\nsemi-locally simply connected, and $\\mathcal{M}_1\\subseteq\\mathcal{M}$. \n\nIf $\\deg f$ is even, then $\\mathcal{V}_f\/\\mathcal{V}_f'$ is isomorphic to the quotient\nof $H_1(\\mathcal{M})$ by the range of the endomorphism $1-\\iota_*\\circ f^!$.\n\nIf $\\deg f$ is odd, then $\\mathcal{V}_f\/\\mathcal{V}_f'$ is isomorphic to the quotient of\n$\\mathbb{Z}\/2\\mathbb{Z}\\oplus H_1(\\mathcal{M})$ by the range of the endomorphism $1-\\sigma_1$,\nwhere $\\sigma_1(t, c)=(t+\\mathop{\\mathrm{sign}}(c), \\iota_*\\circ f^!(c))$.\n\\end{proposition}\n\nHere $\\iota_*:H_1(\\mathcal{M}_1)\\longrightarrow H_1(\\mathcal{M})$ is the homomorphism induced by\nthe identical embedding $\\iota:\\mathcal{M}_1\\longrightarrow\\mathcal{M}$, the homomorphism $f^!:H_1(\\mathcal{M})\\longrightarrow\nH_1(\\mathcal{M}_1)$ maps a cycle $c$ to its full preimage $f^{-1}(c)$, and\n$\\mathop{\\mathrm{sign}}:H_1(\\mathcal{M})\\longrightarrow\\mathbb{Z}\/2\\mathbb{Z}$ maps a cycle $c$ defined by\na loop $\\gamma$ to $1$ if $\\gamma$ acts as an odd permutation on the\nfiber of $f$, and to $0$ otherwise.\n\nThe main result of Section~5 is existence of finite presentation of\n$\\mathcal{V}_f$ when $f$ is expanding. More generally, we show that $\\mathcal{V}_G$ is\nfinitely presented, if $G$ is a \\emph{contracting self-similar group},\nsee Theorem~\\ref{th:finitepresentation}. We also give (in\nSubsection~5.2) a general\ndefinition of the groups $\\mathcal{V}_f$ for expanding maps $f:\\mathcal{M}\\longrightarrow\\mathcal{M}$ (where\n$\\mathcal{M}$ is not required to be path connected). \n\nSection ``Dynamical systems and groupoids'' is an overview of the\ntheory of limit dynamical systems of contracting self-similar groups\nand basic results of hyperbolic groupoids, following~\\cite{nek:book}\nand~\\cite{nek:hyperbolic}. These results are needed for the proof the\nfact that $\\mathcal{V}_f$ is a complete invariant of the dynamical system in\nthe expanding case, which is proved (Theorem~\\ref{th:classification})\nin the last section of the paper.\n\nThe general scheme of the proof of Theorem~\\ref{th:classification}\nis as follows. First, we show, using a theorem of M.~Rubin~\\cite{rubin:reconstr},\nthat two groups $\\mathcal{V}_{f_1}$ and $\\mathcal{V}_{f_2}$ are isomorphic if and only\nif their actions on the corresponding boundaries of trees are\ntopologically conjugate. This implies, that the groupoid of germs $\\mathfrak{G}$\nof\nthe action of the group $\\mathcal{V}_f$ on the boundary of the tree is uniquely\ndetermined by the group $\\mathcal{V}_f$.\n\nThe groupoid $\\mathfrak{G}$ is hyperbolic, and hence it uniquely determines the\nequivalence class of its dual (see~\\cite{nek:hyperbolic}), which is\nthe groupoid generated by the germs of $f:\\mathcal{M}\\longrightarrow\\mathcal{M}$. It remains to\nshow that the dynamical system $f:\\mathcal{M}\\longrightarrow\\mathcal{M}$ is uniquely determined (up\nto topological conjugacy) by the equivalence class of the groupoid of\ngerms generated by it. This is proved using the techniques of\nhyperbolic groupoids. Connectedness of $\\mathcal{M}$ is used in the proof in an\nessential way.\n\nIt is a natural question to ask if Theorem~\\ref{th:classification} is\ntrue in general (without the condition that $\\mathcal{M}$ is path connected). \n\nAs a corollary of the proof of Theorem~\\ref{th:classification}, we\nclarify the relation between the groupoid-theoretic equivalence of\ngroupoids of germs associated with expanding dynamical systems, and\ntheir topological conjugacy, see Theorem~\\ref{th:groupoidequivalence}.\n\n\\subsection*{Acknowledgments}\n\nI am grateful to the organizers of the LMS Durham Symposium\n``Geometric and Cohomological Group Theory'' for inviting me to give a\ntalk, which inspired me to return to the topics of this paper.\n\nThe paper is based in part on work supported by NSF grant DMS1006280.\n\n\\section{Definition of the groups $\\mathcal{V}_f$}\n\n\\subsection{Rooted trees}\n\nLet $T$ be a locally finite rooted tree, and let $v, u$ be its\nvertices. We write $v\\preceq u$ if the path connecting the\nroot to $u$ passes through $v$. This defines a partial order on the\nset of vertices of $T$, and $T$ is its Hasse diagram (though we tend\nto draw rooted trees ``upside down'' with the root on top).\n\nWe denote by $T_v$ the sub-tree with root $v$ spanned by all vertices $u$ such that\n$v\\preceq u$. We have $v\\preceq u$ if and only if $T_v\\supseteq\nT_u$. If $v$ and $u$ are incomparable, then $T_v$ and $T_u$\nare disjoint.\n\n\\begin{figure}\n\\centering\n\\includegraphics{tree.eps}\n\\caption{Rooted tree}\n\\label{fig:tree}\n\\end{figure}\n\n\\emph{Boundary} $\\partial T$ of the tree $T$ is the set of all\ninfinite simple paths starting at the root of $T$. The boundary\n$\\partial T_v$ is naturally identified with the set of paths $w\\in\\partial T$\npassing through $v$. The sets $\\partial T_v$ form a basis of open sets\nof a natural topology on $\\partial T$. The subsets $\\partial T_v$ are clopen (closed and\nopen), and every clopen subset of $\\partial T_v$ is disjoint union\nof a finite number of sets of the form $\\partial T_v$.\n\nThe \\emph{$n$th level} of $T$ is the set of vertices that are on\ndistance $n$ from the root of the tree. \nAn \\emph{antichain} of a rooted tree $T$ is a set of\npairwise incomparable vertices. A finite antichain $A$ is said to be\n\\emph{complete} if it is maximal, i.e., if \nevery set $B$ of vertices of $T$ properly containing $A$\nis not an antichain. For example, every level of $T$ is a complete antichain.\n\nA set $A$ is an antichain if and only if the sets $\\partial T_v$ for\n$v\\in A$ are disjoint. It follows that $A$ is a complete antichain if\nand only if $\\partial T$ is disjoint union of the sets $\\partial T_v$\nfor $v\\in A$.\n\nLet $X$ be a finite set. We denote by $X^*$ the free monoid generated\nby $X$, i.e., the set of all finite words $x_1x_2\\ldots x_n$ for\n$x_i\\in X$, together with the empty word $\\varnothing$.\n\\emph{Length} of a word $v\\in X^*$ is the number of its letters.\n\nWe introduce on the set $X^*$ structure of a rooted tree coinciding with\nthe left Cayley graph of the free monoid. Namely, two finite words are\nconnected by an edge if and only if they are of the form $vx$ and $v$\nfor $v\\in X^*$ and $x\\in X$. The empty word is the root of the tree\n$X^*$.\nWe have $v\\preceq u$ for $v, u\\in X^*$ if and only if $v$ is a\nbeginning of $u$.\nThe subtree $T_v$ of $T=X^*$ for $v\\in X^*$ is the\nset $vX^*$ of all words starting with $v$. The $n$th level of $X^*$ is\nthe set $X^n$ of words of length $n$.\n\nThe boundary of the tree $X^*$ is naturally identified with the space\n$X^\\omega$ of right-infinite sequences $x_1x_2\\ldots$ of elements of\n$X$. The topology on the boundary coincides with the direct product\ntopology on $X^\\omega$.\n\n\\subsection{Definition}\n\nLet $\\mathcal{M}$ be a topological space. A \\emph{partial self-covering} is a finite degree covering map\n$f:\\mathcal{M}_1\\longrightarrow\\mathcal{M}$, where $\\mathcal{M}_1\\subseteq\\mathcal{M}$. \n\nIf $f:\\mathcal{M}_1\\longrightarrow\\mathcal{M}$ is a partial self-covering, then we can iterated it\nas a partial self-map of $\\mathcal{M}$. Then the $n$th iteration\n$f^n:\\mathcal{M}_n\\longrightarrow\\mathcal{M}$ is also a partial self-covering. Here\n$\\mathcal{M}_n\\subset\\mathcal{M}_{n-1}\\subset\\ldots\\subset\\mathcal{M}_1$ are domains of the\niterations of $f$, defined by the condition $\\mathcal{M}_{n+1}=f^{-1}(\\mathcal{M}_n)$.\n\nFor a point $t\\in\\mathcal{M}$ denote by $T_t$ the \\emph{tree of preimages} of\n$t$ under the iterations of $f$, i.e., the tree with the set of\nvertices equal to the formal disjoint union $\\bigsqcup_{n\\ge\n  0}f^{-n}(t)$ of the sets of preimages of $t$ under the iterations\n$f^n:\\mathcal{M}_n\\longrightarrow\\mathcal{M}$. Here $f^{-0}(t)=\\{t\\}$ consists of the root of the\ntree, and a vertex $v\\in f^{-n}(t)$ is connected by an edge to the\nvertex $f(v)\\in f^{-(n-1)}(t)$, see Figure~\\ref{fig:prtree}. \nIf $v$ is a vertex of $T_t$, then the tree of preimages $T_v$ is in a\nnatural way a sub-tree of the tree $T_t$, and our notation agrees with\nthe notation of the previous subsection. \n\n\\begin{figure}\n\\centering\\includegraphics{prtree.eps}\n\\caption{Tree $T_t$}\n\\label{fig:prtree}\n\\end{figure}\n\nAssume now that $\\mathcal{M}$ is path connected. \nLet $t_1, t_2\\in\\mathcal{M}$, and let $\\gamma$ be a path from $t_1$ to $t_2$ in\n$\\mathcal{M}$. Then for every $n\\ge 0$ and every $v\\in f^{-n}(t_1)$ there is a\nunique lift by the covering $f^n:\\mathcal{M}_n\\longrightarrow\\mathcal{M}$ of $\\gamma$ starting in $v$. Let\n$S_\\gamma(v)\\in f^{-n}(t_2)$ be its end. It is easy to see that\nthe map $S_\\gamma:T_{t_1}\\longrightarrow T_{t_2}$ is an isomorphism of the rooted\ntrees, see Figure~\\ref{fig:sgamma}. It defines a homeomorphism of their boundaries\n$S_\\gamma:\\partial T_{t_1}\\longrightarrow\\partial T_{t_2}$, which we will denote by the\nsame letter.\n\n\\begin{figure}\n\\centering\\includegraphics{sgamma.eps}\n\\caption{Isomorphism $S_\\gamma$}\n\\label{fig:sgamma}\n\\end{figure}\n\n\\begin{defi} Let $f:\\mathcal{M}_1\\longrightarrow\\mathcal{M}$ be a partial self-covering, and let $t\\in\\mathcal{M}$.\nDenote by $\\mathcal{T}_f$ the semigroup of partial homeomorphisms of  $\\partial T_t$\ngenerated by the homeomorphisms of the form $S_\\gamma:\\partial T_{v_1}\\longrightarrow\n\\partial T_{v_2}$, where $\\gamma$ is a\npath connecting points $v_1, v_2\\in\\bigcup_{n\\ge 0}f^{-n}(t)$.\n\\end{defi}\n\nThe semigroup $\\mathcal{T}_f$ contains the \\emph{zero} map between empty\nsubsets of $T_t$. A product $S_{\\gamma_1}S_{\\gamma_2}$ is zero if the\nrange of $S_{\\gamma_2}$ is disjoint from the domain of\n$S_{\\gamma_1}$.\n\nLet $S_{\\gamma_1}:\\partial T_{v_1}\\longrightarrow\\partial T_{v_2}$ and\n$S_{\\gamma_2}:\\partial T_{u_1}\\longrightarrow\n\\partial T_{u_2}$ be two generators of $\\mathcal{T}_f$. The product\n$S_{\\gamma_1}S_{\\gamma_2}$ is non-zero if and only if $T_{u_2}$ and\n$T_{v_1}$ are not disjoint, i.e., if either  $T_{u_2}\\supseteq\nT_{v_1}$, or $T_{u_2}\\subseteq\nT_{v_1}$. It the first case, $v_1$ is a\npreimage of $u_2$ under some iteration $f^k$ of $f$. Let $\\gamma_2'$\nbe the unique lift of $\\gamma_2$ by $f^k$ that ends in $v_1$. Then it\nfollows from the definition of the transformations $S_\\gamma$\nthat\n\\begin{equation}\\label{eq:gamma1}\n  S_{\\gamma_1}S_{\\gamma_2}=S_{\\gamma_1\\gamma_2'},\\end{equation}\nsee Figure~\\ref{fig:sg12}. Here and in the sequel, we multiply paths\nas we compose functions: in the product $\\gamma_1\\gamma_2'$ the path\n$\\gamma_2'$ is passed before the path $\\gamma_1$.\n\nSimilarly, if $T_{u_2}\\subseteq T_{v_1}$, then $u_2$ is a\n$f^k$-preimage of $v_1$ for some $k\\ge 0$, and\n\\begin{equation}\\label{eq:gamma2}S_{\\gamma_1}S_{\\gamma_2}=S_{\\gamma_1'\\gamma_2},\\end{equation}\nwhere $\\gamma_1'$ is the lift of $\\gamma_1$ by $f^k$ starting in\n$u_2$.\n\n\\begin{figure}\n\\centering\\includegraphics{sg12.eps}\n\\caption{Composition $S_{\\gamma_1}S_{\\gamma_2}$}\n\\label{fig:sg12}\n\\end{figure}\n\nIt follows that all non-zero elements of $\\mathcal{T}_f$ are of the form\n$S_\\gamma$ for some path $\\gamma$ in $\\mathcal{M}$ connecting two vertices of $T_t$. Note\nalso that $\\mathcal{T}_t$ is an inverse semigroup, where\n$S_\\gamma^*=S_{\\gamma^{-1}}$.\n\nLet $A_1, A_2$ be two complete antichains of $T_t$ of equal cardinality. Choose a\nbijection $\\alpha:A_1\\longrightarrow A_2$ and a collection of paths $\\gamma_a$\nfrom $a\\in A_1$ to the corresponding vertex\n$\\alpha(a)$, see Figure~\\ref{fig:elements}.\nLet $g:\\partial T_t\\longrightarrow\\partial T_t$ be the map given by the rule\n\\[g(w)=S_{\\gamma_v}(w),\\qquad\\text{if $w\\in \\partial T_v$.}\\]\nIt is easy to see that $g:\\partial T_t\\longrightarrow\\partial T_t$ is a\nhomeomorphism. Denote by $\\mathcal{V}_f$ the set of all such homeomorphisms.\n\n\\begin{figure}\n\\centering\n\\includegraphics{elements.eps}\n\\caption{Elements of $\\mathcal{V}_f$}\n\\label{fig:elements}\n\\end{figure}\n\nWe will represent homeomorphisms $g\\in\\mathcal{V}_f$ by tables of the form\n\\begin{equation}\\label{eq:table}\\left(\\begin{array}{cccc}v_1 & v_2 & \\ldots & v_n\\\\ \\gamma_{v_1} &\n    \\gamma_{v_2} & \\ldots & \\gamma_{v_n}\\\\ \\alpha(v_1) & \\alpha(v_2) &\n    \\ldots & \\alpha(v_n)\\end{array}\\right),\\end{equation}\nwhere the first row is the list of vertices of a complete antichain,\nand $g(w)=S_{\\gamma_{v_i}}(w)$ for all $w\\in\\partial T_{v_i}$.\n\nAn \\emph{elementary splitting} of such a table is the operation of replacing a column\n$\\left(\\begin{array}{c} u \\\\ \\gamma_u \\\\ \\alpha(u)\\end{array}\\right)$\nby the array $\\left(\\begin{array}{cccc}u_1 & u_2 & \\ldots & u_d\\\\ \\gamma_{u_1} &\n    \\gamma_{u_2} & \\ldots & \\gamma_{u_d}\\\\ w_1 & w_2 &\n    \\ldots & w_d\\end{array}\\right)$, where $\\{u_1, u_2, \\ldots,\nu_d\\}=f^{-1}(u)$, $\\gamma_{u_i}$ is the lift of $\\gamma_u$ by $f$\nstarting at $u_i$, and $w_i$ is the end of $\\gamma_{u_i}$, see\nFigure~\\ref{fig:splitting}.\nA \\emph{splitting} of a table is the results of a finite sequence of\nelementary splittings.\n\n\\begin{figure}\n\\centering\n\\includegraphics{splitting.eps}\n\\caption{Elementary splitting}\n\\label{fig:splitting}\n\\end{figure}\n\nIt follows directly from the definition that splitting of a table does\nnot change the homeomorphism $g\\in\\mathcal{V}_f$ that it defines.\nIt is obvious that if $g_1$ and\n$g_2$ are defined by tables of the form\n\\[\\left(\\begin{array}{cccc}v_1 & v_2 & \\ldots & v_n\\\\ \\gamma_{v_1} &\n    \\gamma_{v_2} & \\ldots & \\gamma_{v_n}\\\\ u_1 & u_2 & \\ldots &\n    u_n\\end{array}\\right)\\]\nand\n\\[\\left(\\begin{array}{cccc}w_1 & w_2 & \\ldots & w_n\\\\ \\gamma_{w_1} &\n    \\gamma_{w_2} & \\ldots & \\gamma_{w_n}\\\\ v_1 & v_2 & \\ldots &\n    v_n\\end{array}\\right),\\]\nthen the composition $g_1g_2$ is defined by the table\n\\[\\left(\\begin{array}{cccc}w_1 & w_2 & \\ldots & w_n\\\\ \\gamma_{v_1}\\gamma_{w_1} &\n    \\gamma_{v_2}\\gamma_{w_2} & \\ldots & \\gamma_{v_n}\\gamma_{w_n}\\\\ u_1 & u_2 & \\ldots &\n    u_n\\end{array}\\right).\\]\n\nLet \\[\\left(\\begin{array}{cccc}a_1 & a_2 & \\ldots & a_n\\\\ \\gamma_{a_1} &\n    \\gamma_{a_2} & \\ldots & \\gamma_{a_n}\\\\ b_1 & b_2 & \\ldots &\n    b_n\\end{array}\\right),\\quad \\left(\\begin{array}{cccc}c_1 & c_2 & \\ldots & c_m\\\\ \\gamma_{c_1} &\n    \\gamma_{c_2} & \\ldots & \\gamma_{c_m}\\\\ a_1' & a_2' & \\ldots &\n    a_m'\\end{array}\\right)\\] be tables defining elements of $\\mathcal{V}_f$. We can find a\ncomplete antichain $A$ such that for every $v\\in A$ the subtree $T_v$\nis contained in a subtree $T_{a_i}$ and a subtree $T_{a_j'}$ for some\n$i$ and $j$. For\nexample, we can take $A$ to be equal to the $k$th level of the tree\n$T_t$ for $k$ big enough. Then\nthere exists a splitting of the first table such that its first row is\n$A$, and there exists a splitting of the second table such that its\nlast row is $A$.\n\nIt follows that if $g_1, g_2\\in\\mathcal{V}_f$, then their composition $g_1g_2$\nalso belongs to $\\mathcal{V}_f$. As a corollary, we get the following proposition.\n\n\\begin{proposition}\nThe set $\\mathcal{V}_f$ is a group.\n\\end{proposition}\n\nWe will use sometimes instead of tables the following notation.\nIf $F$ and $G$ are two partial transformations with disjoint domains,\nthen we denote by $F+G$ their union, i.e., the map equal to $F$ on the\ndomain of $F$ and equal to $G$ on the domain of $G$. Then the\ntransformation defined by a table\n\\[\\left(\\begin{array}{cccc}v_1 & v_2 & \\ldots & v_n\\\\ \\gamma_1 &\n    \\gamma_2 & \\ldots & \\gamma_n\\\\ u_1 & u_2 & \\ldots &\n    u_n\\end{array}\\right)\\]\nis written $S_{\\gamma_1}+S_{\\gamma_2}+\\cdots+S_{\\gamma_n}$.\n\nAn elementary splitting of a table is equivalent then to application\nof the identity\n\\[S_\\gamma=\\sum_{\\delta\\in f^{-1}(\\gamma)}S_\\delta,\\] where\n$f^{-1}(\\gamma)$ is the set of all lifts of $\\gamma$ by $f$.\n\n\\section{Symbolic coding}\n\\subsection{Two trees}\n\\label{ss:2trees}\n\nLet $\\{t_1, t_2, \\ldots, t_d\\}=f^{-1}(t)$, and choose paths $\\ell_i$ in $\\mathcal{M}$\nfrom $t$ to $t_i$. Then $S_{\\ell_i}:T_t\\longrightarrow T_{t_i}$ is an\nisomorphism. The elements $S_{\\ell_i}\\in\\mathcal{T}_f$ satisfy the\nrelations:\n\\[S_{\\ell_i}^*S_{\\ell_i}=1,\\qquad\\sum_{i=1}^dS_{\\ell_i}S_{\\ell_i}^*=1.\\]\n\nThe $C^*$-algebra defined by such relations is called the \\emph{Cuntz\nalgebra}~\\cite{cuntz}. If we denote by $S$ the row\n$(S_{\\ell_1}, S_{\\ell_2}, \\ldots, S_{\\ell_d})$, and by $S^*$ the\ncolumn $(S_{\\ell_1}^*, S_{\\ell_2}^*, \\ldots,\nS_{\\ell_d}^*)^\\top$, then the relations can be written as matrix equalities\n\\[S^*S=I_d,\\qquad SS^*=I_1,\\]\nwhere $I_n$ denotes the $n\\times n$ identity matrix. Rings given by\nthese and similar defining relations were studied by\nW.~Leavitt~\\cite{leavitt:moduleswords,leavitt:duke}.\n\nLet $\\Gamma_t$ be the graph with the set of vertices equal to the set of\nvertices of $T_t$ in which two vertices $v\\in f^{-n}(t)$ and $v\\in\nf^{-(n+1)}(t)$  are connected by an edge if\nand only if they are connected by a path equal to a lift of a path\n$\\ell_i$ by the covering $f^n$. In other words, the\ngraph $\\Gamma_t$ is obtained by taking preimages of the paths $\\ell_i$\nunder all iterations of $f$, see Figure~\\ref{fig:lambda}.\n\n\\begin{figure}\n\\centering\n\\includegraphics{lambda.eps}\n\\caption{Trees $\\Gamma_t$ and $T_t$}\n\\label{fig:lambda}\n\\end{figure}\n\nIt is easy to see that $\\Gamma_t$ is a tree, and its $n$th level is\nequal to the $n$th level of the tree $T_t$.\nIt follows that any two vertices of $T_t$ are connected by a unique\nsimple path in $\\Gamma_t$.\n\nLet $i_{n-1}\\ldots i_1i_0\\in\\{1, 2, \\ldots, d\\}^n$, and consider the\nproduct $S_{i_{n-1}}\\cdots S_{i_1}S_{i_0}\\in\\mathcal{T}_f$. \nAccording to the multiplication rule~\\eqref{eq:gamma2}, it is equal to\n$S_{\\gamma_{n-1}\\ldots\\gamma_1\\gamma_0}$, where $\\gamma_0=\\ell_{i_0}$, and\n$\\gamma_k$ is the lift of $\\ell_k$ by $f^k$ starting at the end of\n$\\gamma_{k-1}$. Denote by $\\Lambda(i_{n-1}\\ldots i_1i_0)$ the end of\nthe last path $\\gamma_{n-1}$. Then $S_{i_{n-1}}\\cdots\nS_{i_1}S_{i_0}=S_{\\gamma_{i_{n-1}\\ldots i_2i_1}}$, where\n$\\gamma_{i_{n-1}\\ldots i_2i_1}$ is unique simple path in $\\Gamma_t$\nstarting at the root and ending in $\\Lambda(i_{n-1}\\ldots i_1i_0)$.\nThe path $\\gamma_{i_{n-1}\\ldots i_2i_1}$ and its end\n$\\Lambda(i_{n-1}\\ldots i_1i_0)$ satisfy the\nrecurrent rule:\n\\[\\gamma_{i_{n-1}\\ldots i_2i_1}=\\ell_{\\Lambda(i_{n-2}\\ldots i_1i_0),\n  i_{n-1}}\\gamma_{i_{n-2}\\ldots i_2i_1},\\]\nwhere $\\ell_{\\Lambda(i_{n-2}\\ldots i_1i_0), i_{n-1}}$ is the lift of\n$\\ell_{i_{n-1}}$ by $f^{n-1}$ starting at $\\Lambda(i_{n-2}\\ldots\ni_1i_0)$ (and hence ending in $\\Lambda(i_{n-1}\\ldots i_1i_0)$).\n\nThe map $\\Lambda$ is a bijection between $\\{1, 2, \\ldots, d\\}^n$ and the\n$n$th level $f^{-n}(t)$ of the trees $T_t$ and $\\Gamma_t$. It\nfollows directly from the description of the path\n$\\gamma_{i_{n-1}\\ldots i_1i_0}$ that\n$\\Lambda(i_{n-1}\\ldots i_1i_0)$ is adjacent to $\\Lambda(i_{n-1}\\ldots\ni_2i_1)$ in $T_t$ and to $\\Lambda(i_{n-2}\\ldots i_1i_1)$ in $\\Gamma_t$.\nIn other words, $T_t$ and $\\Gamma_t$ are identified by $\\Lambda$ with\nthe right and the left Cayley graphs of the free monoid generated by\n$X=\\{1, 2, \\ldots, d\\}$, respectively.\n\nFor any two sequences $i_1i_2\\ldots i_n$ and $j_1j_2\\ldots j_m\\in\nX^*$, the product \\[S_{\\ell_{i_1}}S_{\\ell_{i_2}}\\cdots\nS_{\\ell_{i_n}}(S_{j_1}S_{j_2}\\ldots S_{j_m})^*\\] is\nequal to $S_\\gamma$, where $\\gamma$ is the path inside $\\Gamma_t$\nfrom $\\Lambda(j_1j_2\\ldots j_m)$ to $\\Lambda(i_1i_2\\ldots i_n)$. This\nfollows directly from the definitions of $\\Gamma_t$, $\\Lambda$, and\nrules~\\eqref{eq:gamma1} and~\\eqref{eq:gamma2}.\n\nWe will use notation $S_x=\\Lambda^{-1}S_{\\ell_x}\\Lambda$ and $S_{x_1x_2\\ldots\n  x_n}=S_{x_1}S_{x_2}\\cdots S_{x_n}$, for $x, x_i\\in X$. Then, by the definition of\n$\\Lambda$, the transformations  $S_{x_1x_2\\ldots x_n}$ of $X^\\omega$ are given by the rule\n\\[S_{x_1x_2\\ldots x_n}(v)=x_1x_2\\ldots x_nv.\\]\nFor $v, u\\in X^*$, the transformation $S_vS_u^*$ is defined on\n$uX^\\omega$, and acts by the rule\n\\[S_vS_u^*(vw)=uw.\\]\nIn particular, $\\sum_{x\\in X}S_xS_x^*=1$, and we obviously have $S_x^*S_x=1$.\n\n\\subsection{The Higman-Thompson group}\n\nLet $A_1$ and $A_2$ be complete antichains in $X^*$, and let\n$\\alpha:A_1\\longrightarrow A_2$ be a bijection. Then $g_\\alpha=\\sum_{v\\in\n  A_1}S_{\\alpha(v)}S_v^*$ is a homeomorphism of $X^\\omega$ defined\nby the rule\n\\[g_\\alpha(vw)=\\alpha(v)w,\\]\nfor all $v\\in A_1$ and $w\\in X^\\omega$. The set of such homeomorphisms\n$g_\\alpha$ is the \\emph{Higman-Thompson group} group $G_{|X|, 1}$, see~\\cite{hgthomp}, which we\nwill denote by $\\mathcal{V}_X$ of $\\mathcal{V}_d$, where $d=|X|$. \n\nIts copy $\\Lambda\\cdot\\mathcal{V}_X\\cdot\\Lambda^{-1}$ in $\\mathcal{V}_f$ is the\ngroup defined by the paths in the tree $\\Gamma_t$. Namely, for any bijection\n$\\alpha:A_1\\longrightarrow A_2$ between complete antichains of $T_t$ there exist\nunique simple paths $\\gamma_v$ connecting $v\\in A_1$ to\n$\\alpha(v)\\in A_2$ inside the tree $\\Gamma_t$. Then the corresponding\nelement of $\\mathcal{V}_f$ is equal to $\\sum_{v\\in A_1}S_{\\gamma_v}$.\n\nThe following simple lemma will be useful later (for a proof, see, for\nexample~\\cite[Lemma~9.12]{nek:bim}).\n\n\\begin{lemma}\n\\label{lem:incomplete}\nLet $A_1, A_2\\subset X^*$ be finite incomplete (i.e., non-maximal)\nantichains, and let $\\alpha:A_1\\longrightarrow A_2$ be a bijection. Then there\nexists $g\\in\\mathcal{V}_X$ such that $g(vw)=\\alpha(v)w$ for all $v\\in A_1$ and\n$w\\in X^\\omega$.\n\\end{lemma}\n\n\n\\subsection{The iterated monodromy group}\n\nEvery element $\\gamma$ of the fundamental group $\\pi_1(\\mathcal{M}, t)$ \ndefines an element $S_\\gamma:\\partial T_t\\longrightarrow\\partial T_t$ of $\\mathcal{V}_f$. We get in\nthis way a natural homomorphism $\\gamma\\mapsto S_\\gamma$ from\n$\\pi_1(\\mathcal{M}, t)$ to $\\mathcal{V}_f$. Its image is called the \\emph{iterated\n  monodromy group} of $f$ and is denoted $\\img{f}$. It acts on $T_t$\nby automorphisms, so that the action on the $n$th level coincides with\nthe natural \\emph{monodromy action} associated with the covering\n$f^n:\\mathcal{M}_n\\longrightarrow\\mathcal{M}$, see~\\cite[Chapter~5]{nek:book},~\\cite{bgn,nek:bath}.\n\nLet us choose paths $\\ell_i$ connecting the root $t$ to the vertices\nof the first level $f^{-1}(t)$ of the tree $T_t$. Let $\\Gamma_t$ be the\ntree obtained by taking lifts of the paths $\\ell_i$ by iterations of\n$f$, as in Subsection~\\ref{ss:2trees}.\n\nFor a vertex $v$ of $T_t$, denote by $\\ell_v$ the unique simple path\ninside $\\Gamma_t$ from $t$ to $v$.\nThen for an arbitrary path $\\gamma$ in $\\mathcal{M}$ starting in a vertex $v$ and ending in\na vertex $u$ of $T_t$, the path $\\ell_u^{-1}\\gamma\\ell_v$ is a loop\nbased at $t$. Let $g=S_{\\ell_u^{-1}\\gamma\\ell_v}$ be the corresponding\nelement of $\\img{f}$. Then we have\n\\[S_\\gamma=S_{\\ell_u}S_{\\ell_u^{-1}\\gamma\\ell_v}S_{\\ell_v}^*=S_{\\ell_u}gS_{\\ell_v}^*.\\]\nHence we get the following description of the elements of $\\mathcal{V}_f$.\n\\begin{lemma}\n\\label{lem:imginside}\nLet $g\\in\\mathcal{V}_f$ be defined by a table \n$\\left(\\begin{array}{cccc}\n   v_1 & v_2 & \\ldots & v_n\\\\\n   \\gamma_1     & \\gamma_2     & \\ldots & \\gamma_n \\\\\n   u_1 & u_2 & \\ldots & u_n\n\\end{array}\\right)$. Denote\n$g_i=S_{\\ell_{u_i}}^*S_{\\gamma_i}S_{\\ell_{v_i}}$. Then\n$g_i\\in\\img{f}$, and\n$g=\\sum_{i=1}^n S_{\\ell_{u_i}}g_iS_{\\ell_{v_i}}^*$.\n\\end{lemma}\n\nLet $g\\in\\img{f}$ be defined by a loop $\\gamma$, and let\n$x\\in f^{-1}(t)$ be a vertex of the first level. Let $\\gamma_x$ be the\nlift of $\\gamma$ by $f$ starting at $x$, and let $y$ be its end. Then\nwe have\n\\begin{equation}\\label{eq:ssimilarity}\ngS_{\\ell_x}=S_\\gamma S_{\\ell_x}=S_{\\gamma_x\\ell_x}=\nS_{\\ell_y\\ell_y^{-1}\\gamma_x\\ell_x}=S_{\\ell_y}S_{\\ell_y^{-1}\\gamma_x\\ell_x}.\n\\end{equation}\nNote that $\\ell_y^{-1}\\gamma_x\\ell_x$ is a loop based at $t$, i.e., an\nelement of $\\pi_1(\\mathcal{M}, t)$, see Figure~\\ref{fig:recurn}. \n\n\\begin{figure}\n\\centering\n\\includegraphics{recurn.eps}\n\\caption{Iterated monodromy recursion}\n\\label{fig:recurn}\n\\end{figure}\n\nLet us conjugate the action of $\\mathcal{V}_f$ on $\\partial T_t$ (and the\naction of $\\img{f}$ on $T_t$) to an action on $X^\\omega$ (and $X^*$)\nusing the isomorphism $\\Lambda:X^*\\longrightarrow T_t$. We call the obtained\nactions of $\\mathcal{V}_f$ and $\\img{f}$ \\emph{standard}. Then\nformula~\\eqref{eq:ssimilarity}\nproves the following lemma, see also~\\cite[Proposition~5.2.2]{nek:book}.\n\n\\begin{lemma}\n\\label{lem:ssimilarity}\nFor every $g\\in\\img{f}$ and every $x\\in f^{-1}$ there exist\n$h\\in\\img{f}$ and $y\\in f^{-1}$ such that\n\\[gS_x=S_yh.\\]\nMoreover, if $g$ is defined by a loop $\\gamma$, then $h$ is defined by\nthe loop $\\ell_y^{-1}\\gamma_x\\ell_x$, where $\\gamma_x$ is the lift of\n$\\gamma$ by $f$ starting at $x$.\n\\end{lemma}\n\nIf $gS_x=S_yh$ for $g, h\\in\\img{f}$ and $x, y\\in X$, then we\ndenote $h=g|_x$ and $y=g(x)$. Note that the last equality agrees with\nthe definition of $g$ as an automorphism $S_\\gamma$ of the tree $T_t$.\n\nSince $1=\\sum_{x\\in X}S_xS_x^*$, we have\n\\begin{equation}\n\\label{eq:gsplitting}\ng=\\sum_{x\\in X}gS_xS_x^*=\\sum_{x\\in X}S_{g(x)}g|_xS_x^*,\n\\end{equation}\nwhich gives us a splitting rule for the expressions for elements of\n$\\mathcal{V}_f$ given in Lemma~\\ref{lem:imginside}.\n\nNamely, we get the following description of $\\mathcal{V}_f$ in terms of\n$\\img{f}$ and formula~\\eqref{eq:gsplitting}.\n\n\\begin{proposition}\n\\label{pr:symbolic}\nThe group $\\mathcal{V}_f$ is isomorphic to homeomorphisms of $X^\\omega$ of the form\n\\begin{equation}\\label{eq:SugSv}\n\\sum_{v\\in A_1}S_{\\alpha(v)}g_vS_v^*,\n\\end{equation}\nwhere $g_v\\in\\img{f}$, $A_1$ is a complete antichain in $X^*$, and\n$\\alpha:A_1\\longrightarrow A_2$ is a bijection of $A_1$ with a complete antichain\n$A_2$. Two elements of $\\mathcal{V}_f$ given by expressions of the form~\\eqref{eq:SugSv}\nare equal if and only if they can be made equal after repeated\napplications of the splitting rules~\\eqref{eq:gsplitting} to the elements\n$g_v$.\n\\end{proposition}\n\nEquivalently, we can use the table notation, and represent the\nelement~\\eqref{eq:SugSv} by the table\n\\begin{equation}\n\\label{eq:tablesG}\n\\left(\\begin{array}{cccc} v_1 & v_2 & \\ldots & v_m\\\\ g_{v_1} &\n    g_{v_2} & \\ldots & g_{v_m}\\\\ \\alpha(v_1) & \\alpha(v_2) & \\ldots &\n    \\alpha(v_m)\\end{array}\\right),\n\\end{equation}\nwhere the splitting rule is the operation of replacing a column by the\narray\n\\begin{equation}\n\\label{eq:tablesspittingG}\n\\left(\\begin{array}{c}v \\\\ g\\\\ u\\end{array}\\right)\\mapsto\\left(\\begin{array}{cccc}vx_1 & vx_2 & \\ldots & vx_d\\\\ g|_{x_1} &\n    g|_{x_2} & \\ldots & g|_{x_d}\\\\ ug(x_1) & ug(x_2) & \\ldots &\n    ug(x_d)\\end{array}\\right).\n\\end{equation}\n\n\\begin{example}\n\\label{ex:addingmachine}\nConsider the self-covering $f:x\\mapsto 2x$ of the circle $\\mathbb{R}\/\\mathbb{Z}$. Take\n$t=0$ as the basepoint. Its preimages are $0$ and $1\/2$. Let $\\ell_0$\nbe the trivial path at $0$, and let $\\ell_1$ be the path from $0$ to\n$1\/2$ equal to the image of the segment $[0, 1\/2]\\subset\\mathbb{R}$. Let $\\gamma$\nbe the generator of $\\pi_1(\\mathbb{R}\/\\mathbb{Z}, 0)$ equal to the image of the\nsegment $[0, 1]\\subset\\mathbb{R}$ with the natural (increasing on $[0, 1]$)\norientation. It has two lifts by the covering $f$: \\[\\gamma_0=[0,\n1\/2],\\qquad\\gamma_1=[1\/2, 1].\\]\nNote that $\\gamma_0=\\ell_1$.\n\nBy~\\eqref{eq:gamma2}, \n\\begin{equation}\n\\label{eq:admach1}\nS_\\gamma S_{\\ell_0}=S_{\\gamma_0\\ell_0}=S_{\\ell_1},\n\\end{equation}\nand\n\\begin{equation}\n\\label{eq:admach2}\nS_\\gamma\nS_{\\ell_1}=S_{\\gamma_1\\ell_1}=S_{\\ell_0\\gamma}=S_{\\ell_0}S_{\\gamma}.\n\\end{equation}\n\nLet $X=\\{0, 1\\}$, and consider the corresponding standard actions on\n$X^*$ and $X^\\omega$. Denote by $a$ the generator of $\\img{f}$\ncorresponding to $S_\\gamma$. Then, by~\\eqref{eq:admach1} and~\\eqref{eq:admach2},\n\\[aS_0=S_1,\\qquad aS_1=S_0a.\\]\nIn other words, the action of $a$ on $X^\\omega$ is given by\nthe recurrent formulas\n\\[a(0v)=1v,\\qquad a(1v)=0a(v).\\]\nWe see that $a$ acts as the \\emph{binary adding machine}, see~\\cite[Section~1.7.1]{nek:book}.\n\n\\begin{figure}\n\\centering\n\\includegraphics{splittinga.eps}\n\\caption{The adding machine}\n\\label{fig:splittinga}\n\\end{figure}\n\nThe group $\\mathcal{V}_f$ is generated by the Higman-Thompson group $\\mathcal{V}_2$\n(also coinciding with the Thompson group $V$, see~\\cite{intro_tomp}) and an element\n$a$ satisfying the splitting rule $a=S_1S_0^*+S_0aS_1^*$, i.e.,\n\\[\\left(\\begin{array}{c}v\\\\ a\\\\ u\\end{array}\\right)=\n\\left(\\begin{array}{cc}v0 & v1\\\\ 1 & a\\\\ u1 & u0\\end{array}\\right).\\]\nSee Figure~\\ref{fig:splittinga}.\n\\end{example}\n\n\\begin{example}\n\\label{ex:basilica}\nConsider the complex polynomial $z^2-1$ as a partial self-covering\n$f:\\mathcal{M}_1\\longrightarrow\\mathcal{M}$, where $\\mathcal{M}=\\mathbb{C}\\setminus\\{0, -1\\}$, and\n$\\mathcal{M}_1=\\mathbb{C}\\setminus\\{0, \\pm 1\\}$. Alternatively, we can consider it as a\nself-covering of its Julia set, see Figure~\\ref{fig:basilica}.\n\n\\begin{figure}\n\\centering\n\\includegraphics{basilica.eps}\n\\caption{Julia set of $z^2-1$}\n\\label{fig:basilica}\n\\end{figure}\n\nThen $\\mathcal{V}_{z^2-1}$ is generated by the Thompson group $\\mathcal{V}_2$ and two\nelements $a, b$ satisfying\n\\[a=S_1S_0^*+S_0bS_1^*,\\qquad b=S_0S_0^*+S_1aS_1^*.\\]\n\nFor a detailed proof of the recurrent definitions of the elements $a$\nand $b$, see~\\cite[Subsection~5.2.2.]{nek:book}.\n\\end{example}\n\n\\subsection{Self-similar groups}\n\nThe description of $\\mathcal{V}_f$ in terms of the iterated monodromy\ngroup given in Proposition~\\ref{pr:symbolic}\ncan be generalized in the following way.\n\n\\begin{defi}\n\\label{def:ssimilar}\nLet $G$ be a group acting faithfully by automorphisms of the tree\n$X^*$. We say that $G$ is a \\emph{self-similar group} if for all $g\\in G$ and\n$x\\in X$ there exist $h\\in G$ and $y\\in X$ such that\n\\[g(xw)=yh(w)\\]\nfor all $w\\in X^*$.\n\\end{defi}\n\nWe will usually denote self-similar groups as pairs $(G, X)$.\nNote that the equation in Definition~\\ref{def:ssimilar} is equivalent\nto the equality\n\\[g\\cdot S_x=S_y\\cdot h,\\]\nof compositions of self-maps of $X^\\omega$,\nwhere $S_x$ is, as before, the transformation $S_x(w)=xw$ of\n$X^\\omega$.\n\n\\begin{defi}\n\\label{def:VG}\nLet $G$ be a self-similar group acting on $X^*$. The group $\\mathcal{V}_G$ is\nthe set of all homeomorphisms $g$ of $X^\\omega$ for which there exist\ncomplete antichains $A_1, A_2\\subset X^*$, a bijection $\\alpha:A_1\\longrightarrow\nA_2$, and elements $g_v\\in G$, for $v\\in A_1$ such that\n\\[g=\\sum_{v\\in A_1}S_{\\alpha(v)}g_vS_v^*,\\]\ni.e.,\n\\[g(vw)=\\alpha(v)g_v(w)\\]\nfor all $v\\in A_1$ and $w\\in X^\\omega$.\n\\end{defi}\n\nIf $G$ is a self-similar group acting on $X^*$, then for every $v\\in\nX^*$ there exists a unique element of $G$, denoted $g|_v$, such that\n\\[g(vw)=g(v)g|_v(w)\\]\nfor all $w\\in X^\\omega$. We call $g|_v$ the \\emph{section} of $g$ in\n$v$. \n\nThe elements of $\\mathcal{V}_G$ are represented by tables of the\nform~\\eqref{eq:tablesG} with the splitting\nrule~\\eqref{eq:tablesspittingG}. The following proposition follows now\ndirectly from the described constructions.\n\n\\begin{proposition}\nConsider $\\img{f}$ as a self-similar group with respect to a standard\naction on $X^*$. Then $\\mathcal{V}_f$ (with the corresponding standard action\non $X^\\omega$) is equal to $\\mathcal{V}_{\\img{f}}$.\n\\end{proposition}\n\nLet us describe more examples of groups $\\mathcal{V}_f$ and $\\mathcal{V}_G$.\n\n\\begin{example}\n\\label{ex:GrigorchukG}\nThe Grigorchuk group $G$ is generated by the transformations\n\\[a=S_1S_0^*+S_0S_1^*,\\quad b=S_0aS_0^*+S_1cS_1^*,\nc=S_0aS_0^*+S_1dS_1^*, d=S_0S_0^*+S_1bS_1^*,\\]\nsee~\\cite{grigorchuk:80_en}.\n\nThe corresponding group $\\mathcal{V}_G$ was defined and studied \nby C.~R\\\"over in~\\cite{roever,roever:comm}. This was the first example\nof a group $\\mathcal{V}_G$.\nC.~R\\\"over proved that $\\mathcal{V}_G$ is isomorphic to the\nabstract commensurator of $G$, that it is finitely presented, and\nsimple. We will study the last two properties of the groups $\\mathcal{V}_G$,\ngeneralizing the results of C.~R\\\"over for a wide class of self-similar groups.\n\\end{example}\n\n\n\\begin{example}\n\\label{ex:kneadingv}\nLet $f(z)=z^2+c$ be a complex quadratic polynomial such that\n$f^n(0)=0$ for some $n$ (we assume that $n$ is the smallest number\nwith this property). Then $f$ is a self-covering of its Julia set,\nwhich is path connected. The iterated monodromy groups $\\img{f}$\nassociated with such polynomials were described\nin~\\cite{bartnek:mand}.\nThere exists a\nsequence $v=x_1x_2\\ldots x_{n-1}\\in\\{0, 1\\}^{n-1}$ such that $\\img{f}$ is isomorphic to\nthe group $\\mathfrak{K}_v$ generated by $n$ elements $a_0, a_1,\n\\ldots, a_{n-1}$ given by the recurrent relations\n\\[a_0=S_1S_0^*+S_0a_{n-1}S_1^*,\\]\nand\n\\[a_i=\\left\\{\\begin{array}{cc} S_0a_{i-1}S_0^*+S_1S_1^* & \\text{if $x_i=0$}\\\\\nS_0S_0^*+S_1a_{i-1}S_1^* & \\text{if $x_i=1$},\\end{array}\\right.\\]\nfor $i=1, 2, \\ldots, n-1$. For example, $\\img{z^2-1}=\\mathfrak{K}_0$.\n\\end{example}\n\n\\subsection{Wreath recursions}\n\nLet $(G, X)$ be a self-similar group. Every element $g\\in G$ defines a\npermutation $\\sigma_g$ of $X=X^1\\subset X^*$, and an element of $G^X$\nequal to the function $f_g:x\\mapsto g|_x$. It is easy to check that \nthe map $\\psi:G\\longrightarrow \\symm{X}\\ltimes G^X$ mapping $g$ to\n$(\\sigma_g, f_g)$ is a homomorphism of groups, which we call the\n\\emph{wreath recursion} associated with the self-similar group.\n\nLet $X=\\{1, 2, \\ldots, d\\}$. We will write elements of\n$\\symm{d}\\ltimes G^d=\\symm{X}\\ltimes G^X$ as products\n$\\sigma(g_1, g_2, \\ldots, g_d)$, where $\\sigma\\in\\symm{d}$ and $(g_1,\ng_2, \\ldots, g_d)\\in G^d$.\nMultiplication rule for elements of the wreath\nproduct $G\\wr\\symm{d}=\\symm{d}\\ltimes G^d$ is given by the formula\n\\begin{equation}\n\\label{eq:multiplicationwreath}\n\\sigma(g_1, g_2, \\ldots, g_d)\\pi(h_1, h_2, \\ldots,\nh_d)=\\sigma\\pi(g_{\\pi(1)}h_1, g_{\\pi(2)}h_2, \\ldots, g_{\\pi(d)}h_d).\n\\end{equation}\n\nThe wreath recursion completely describes the self-similar group $G$\nby giving recurrent formulas for the action of its elements on $X^*$.\n\n\\begin{example}\nThe adding machine, see Example~\\ref{ex:addingmachine},\nis given by the recursion $a=\\sigma(1, a)$, where\n$\\sigma$ is the transposition $(0, 1)$. The generators of\n$\\img{z^2-1}$ are given by\n\\[a=\\sigma(1, b),\\qquad b=(1, a),\\]\nsee Example~\\ref{ex:basilica}.\n\\end{example}\n\nAny homomorphism\n$\\psi:G\\longrightarrow\\symm{d}\\ltimes G^d$ defines an action of $G$ on $X^*$ (for\n$X=\\{1, 2, \\ldots, d\\}$) by\nthe recurrent rule:\n\\[g(xw)=\\sigma(x)g_x(w),\\]\nwhere $\\sigma$ and $g_x$ are defined by the condition\n$\\psi(g)=\\sigma(g_1, g_2, \\ldots, g_d)$. This action is not\nfaithful in general. The quotient of $G$ by the kernel of its action\non $X^*$ is called the \\emph{faithful quotient} of $G$, and it is a\nself-similar group in the sense of Definition~\\ref{def:ssimilar}.\n\nLet $\\psi:G\\longrightarrow\\symm{d}\\ltimes G^d$ be an arbitrary homomorphism. Then\nwe can define the group $\\mathcal{V}_\\psi$ associated with it in the same way\nas the groups $\\mathcal{V}_G$ were defined for self-similar groups. Namely,\nelements of $\\mathcal{V}_\\psi$ are defined by tables of the form\n\\[\\left(\\begin{array}{cccc} v_1 & v_2 & \\ldots & v_n\\\\ g_1 & g_2 &\n    \\ldots & g_n\\\\ u_1 & u_2 & \\ldots & u_n\\end{array}\\right),\\]\nwhere $g_i\\in G$, and $\\{v_1, v_2, \\ldots, v_n\\}$ and $\\{u_1, u_2,\n\\ldots, u_n\\}$ are complete antichains of $X^*$. Two tables define the\nsame element if they can be made equal (up to permutations of the\ncolumns) by iterated replacement of a\ncolumn $\\left(\\begin{array}{c}v\\\\ g\\\\ u\\end{array}\\right)$\nby the columns\n\\[\\left(\\begin{array}{cccc} v1 & v2 & \\ldots & vd\\\\ g_1 & g_2 & \\ldots &\n    g_d\\\\\nu\\sigma(1) & u\\sigma(2) & \\ldots & u\\sigma(d)\\end{array}\\right),\\]\nwhere $\\psi(g)=\\sigma(g_1, g_2, \\ldots, g_d)$.\nMultiplication of the tables is defined by the rule\n\\[\\left(\\begin{array}{ccc} w_1 & \\ldots & w_n\\\\ g_1 &\n    \\ldots & g_n\\\\ u_1 & \\ldots & u_n\\end{array}\\right)\n\\left(\\begin{array}{cccc} v_1 & \\ldots & v_n\\\\ h_1 & \\ldots & h_n\\\\ w_1\n    & \\ldots & w_n\\end{array}\\right)=\n\\left(\\begin{array}{ccc} v_1 & \\ldots & v_n\\\\ g_1h_1 & \\ldots & g_nh_n\\\\\n    u_1 & \\ldots & u_n\\end{array}\\right).\\] \n\n\\subsection{Bisets}\nA formalism equivalent to wreath recursions is provided by the notion of a\n\\emph{covering biset}.\nIf $(G, X)$ is a self-similar group, then \nthe set of transformations $S_xg:w\\mapsto xg(w)$ of $X^\\omega$ is\ninvariant under the left and right multiplications by elements of $G$:\n\\[S_xg\\cdot h=S_x(gh),\\qquad h\\cdot S_xg=S_{h(x)}(h|_xg).\\]\nWe get therefore commuting left and right actions of $G$ on the set\n$\\Phi=\\{S_xg\\;:\\;x\\in X, g\\in G\\}$. \nWe will write elements $S_x\\cdot g$ of $\\Phi$ just as $x\\cdot g$.\n\nWe adopt the following definition.\n\n\\begin{defi}\nLet $G$ be a group. A \\emph{$G$-biset} is a set $\\Phi$ together with\ncommuting left and right $G$-actions. It is called a \\emph{covering\nbiset} if the right action is free (i.e., if $x\\cdot g=x$ for\n$x\\in\\Phi$ and $g\\in G$ implies $g=1$) and has a finite number of\norbits.\n\\end{defi}\n\nLet $\\Phi_1, \\Phi_2$ be $G$-bisets. Then their tensor product\n$\\Phi_1\\otimes\\Phi_2$ is defined as the quotient of the set\n$\\Phi_1\\times\\Phi_2$ by the identifications\n\\[(x\\cdot g)\\otimes y=x\\otimes (g\\cdot y),\\qquad g\\in G.\\]\n\nLet $\\Phi$ be the biset $\\{x\\cdot g\\;:\\;x\\in X, g\\in\nG\\}$ associated with a self-similar group. Then every element\nof $\\Phi^{\\otimes n}$ can be uniquely written in the form $x_1\\otimes\nx_2\\otimes\\cdots\\otimes x_n\\cdot g$, where $x_i=x_i\\cdot 1$ are\nelements of $X$. It follows that $n$th tensor power\n$\\Phi^{\\otimes n}$ is naturally identified with the set of pairs\n$v\\cdot g$, for $v\\in X^n$ and $g\\in G$, with the actions\n\\[h\\cdot (v\\cdot g)=h(v)\\cdot (h|_vg),\\qquad (v\\cdot g)\\cdot h=v\\cdot (gh).\\]\n\nLet $\\Phi$ be an arbitrary covering $G$-biset. Choose a transversal\n$X\\subset\\Phi$ of the orbits of the right action. Then every element\nof $\\Phi$ is uniquely written in the form $x\\cdot g$ for $x\\in X$ and\n$g\\in G$. For every $g\\in G$ and $x\\in X$ there exist $h\\in G$ and\n$y\\in X$ such that \\[g\\cdot x=y\\cdot h,\\]\nand the elements $y, h$ are uniquely determined by $g$ and $x$. We get\nhence a homomorphism $\\psi:G\\longrightarrow \\symm{X}\\ltimes G^X$, called the\n\\emph{wreath recursion} associated with $\\Phi$ and $X$. Namely,\n$\\psi(g)=\\sigma\\cdot f$, where $\\sigma\\in\\symm{X}$ and $f\\in G^X$ satisfy\n\\[g\\cdot x=\\sigma(x)\\cdot f(x)\\]\nfor all $x\\in X$ (where\n$G^X$ is seen as the set of functions $X\\longrightarrow G$). If we change the\norbit transversal $X$ to an orbit transversal $Y$, then the\nhomomorphism $\\psi:G\\longrightarrow\\symm{|X|}\\ltimes G^{|X|}$ is composed with an\ninner automorphism of the wreath product (after we identify $X$\nwith $Y$ by a bijection).\nNote that the biset $\\Phi$ is uniquely determined, up to an\nisomorphism of biset, by the homomorphism $\\psi$.\n\n\\begin{defi}\n\\label{def:ssequivalent}\nTwo self-similar actions $(G, X_1)$ and $(G, X_2)$ of a group $G$ are\ncalled \\emph{equivalent} if their associated bisets $\\Phi_i=X_i\\cdot\nG$ are isomorphic, i.e., if there exists a bijection\n$F:\\Phi_1\\longrightarrow\\Phi_2$ such that $F(g_1\\cdot a\\cdot g_2)=g_1\\cdot\nF(a)\\cdot g_2$ for all $g_1, g_2\\in G$ and $a\\in\\Phi_1$. Two\nself-similar actions of groups $G_1, G_2$ are equivalent if they\nbecome equivalent after identification of the groups $G_1, G_2$ by an\nisomorphism $G_1\\longrightarrow G_2$.\n\\end{defi}\n\nThe wreath recursion can be defined invariantly, without a choice\nof the orbit transversal. Namely, let $\\mathop{\\mathrm{Aut}}(\\Phi_G)$\nbe the automorphism group of the right $G$-set $\\Phi$, i.e., the set\nof all bijections $\\alpha:\\Phi\\longrightarrow\\Phi$ such that $\\alpha(x\\cdot\ng)=\\alpha(x)\\cdot g$. Then $\\mathop{\\mathrm{Aut}}(\\Phi_G)$ is\nisomorphic to the wreath product $\\symm{d}\\ltimes G^d$, where $d$ is the\nnumber of the orbits of the right action on $\\Phi$, since the right\n$G$-set $\\Phi$ is free and has $d$ orbits, i.e., is isomorphic to the\ndisjoint union of $d$ copies of $G$. For every element\n$g\\in G$ the map $\\psi(g):x\\mapsto g\\cdot x$ is an automorphism of the right\n$G$-set $\\Phi$. Then $\\psi:G\\longrightarrow\\mathop{\\mathrm{Aut}}(\\Phi_G)$ is the\nwreath recursion. For more on wreath recursions and bisets, \nsee~\\cite{nek:book,nek:filling,nek:models}.\n\n\\begin{example} Let $f:\\mathcal{M}_1\\longrightarrow\\mathcal{M}$ be a self-covering map, and let\n  $\\iota:\\mathcal{M}_1\\longrightarrow\\mathcal{M}$ be a continuous map (for example, $f$ is a\n  partial self-covering, and $\\iota$ is the identical embedding).\n\nSuppose that $\\mathcal{M}$ is path-connected. Choose a basepoint $t\\in\\mathcal{M}$, and \nconsider the set $\\Phi$ of pairs $(z, \\ell)$, where $z\\in f^{-1}(t)$,\nand $\\ell$ is a homotopy class of a path in $\\mathcal{M}$ from $t$ to\n$\\iota(z)$. Then $\\pi_1(\\mathcal{M}, t)$ acts on $\\mathcal{M}$ be appending loops to the\nbeginning of the path $\\ell$:\n\\[(z, \\ell)\\cdot\\gamma=(z, \\ell\\gamma).\\]\nIt also acts by appending images of lifts of $\\gamma$ to the end of\nthe path $\\ell$:\n\\[\\gamma(z, \\ell)=(z', \\iota(\\gamma_z)\\ell),\\]\nwhere $\\gamma_z$ is the lift of $\\gamma$ by $f$ starting at $z$, and\n$z'$ is the end of $\\gamma_z$. Here, as before, we multiply paths as\nfunctions (second path in a product is passed first).\n\nThen $\\Phi$ is a covering $\\pi_1(\\mathcal{M}, t)$-biset. The associated wreath\nrecursion coincides with the wreath recursion associated with the\nstandard action of $\\img{f}$.\n\\end{example}\n\nLet us show a more canonical definition of the groups $\\mathcal{V}_\\psi$ in terms\nof bisets.\nLet $\\Phi$ be a covering $G$-biset. Consider the biset $\\Phi^*$ equal\nto the disjoint union of the bisets $\\Phi^{\\otimes n}$ for all\nintegers $n\\ge 0$. Here $\\Phi^{\\otimes 0}$ is the group $G$ with the\nnatural $G$-biset structure. The set $\\Phi^*$ is a semigroup with respect to the\ntensor product operation.\n\nLet us order the semigroup $\\Phi^*$ with respect to the left divisibility,\ni.e., $v\\preceq u$ if and only if there exists $w$ such that\n$u=v\\otimes w$. It is easy to check that $\\Phi^*$ is\nleft-cancellative, i.e., that $v\\otimes w_1=v\\otimes w_2$ implies that\n$w_1=w_2$.\n\nThe quotient of $\\Phi^*$ by the right $G$-action is a rooted\n$d$-regular tree, and the image of $\\preceq$ under the quotient map is the natural order\non the rooted tree $\\Phi^*\/G$. If $X$ is a right orbit transversal of\n$\\Phi$, then $X^{\\otimes n}=X^n$ is a right orbit transversal of\n$\\Phi^{\\otimes n}$, and the identical embedding of $X^*=\\bigcup_{n\\ge\n  0}X^{\\otimes n}$ into $\\Phi^*$\ninduces an isomorphism of the rooted tree $X^*$ with\n$\\Phi^*\/G$.\n\nThe left action of $G$ on $\\Phi^*$ permutes the orbits of the right\naction and preserves the relation $\\preceq$, hence $G$ acts on the tree $\\Phi^*\/G$ by\nautomorphisms. The corresponding action on $X^*$ is the self-similar\naction defined by the wreath recursion associated with $X$.\n\nLet $A_1, A_2\\subset\\Phi^*$ be finite maximal\nantichains with respect to the divisibility order $\\preceq$. Note that\na subset $A\\subset\\Phi^*$ is a finite maximal antichain if and only if\nits image in $\\Phi^*\/G$ is a maximal antichain. Choose a\nbijection $\\alpha:A_1\\longrightarrow A_2$. \n\nIf $w\\in\\Phi^*$ is such that $v\\preceq\nw$ for some $v\\in A_1$, then there exists a unique $u\\in\\Phi^*$ such\nthat $w=v\\otimes u$. Consider then the transformation $h_{A_1, \\alpha,\n  A_2}:w\\mapsto\n\\alpha(v)\\otimes u$. The map $h_{A_1, \\alpha, A_2}$ is defined for all elements of\n$\\Phi^*$ bigger than some element of $A_1$, hence for all elements of\n$\\Phi^{\\otimes n}$, where $n$ is big enough. We will identify to\ntransformation $h_{A_1, \\alpha, A_2}$ and $h_{A_1', \\alpha', A_2'}$ if\ntheir actions on the sets $\\Phi^{\\otimes n}$ agree for all $n$ big\nenough. It is not hard to prove that the set of equivalence classes of\nsuch maps is a group, which we will denote $\\mathcal{V}_\\Phi$. It is also\nstraightforward to show that $\\mathcal{V}_\\Phi$ coincides with $\\mathcal{V}_\\psi$, where\n$\\psi$ is the wreath recursion associated with $\\Phi$, and that \nif $\\Phi$ is the usual biset associated\nwith a self-similar group $G$, then $\\mathcal{V}_\\Phi$ coincides with $\\mathcal{V}_G$.\n\n\\subsection{Epimorphism onto the faithful quotient}\n\nConsider a covering $G$-biset $\\Phi$ and the corresponding group $\\mathcal{V}_\\Phi$.\nThe faithful quotient $\\overline G$ is a self-similar group acting on $X^*$. We\nhave, therefore two groups: $\\mathcal{V}_\\Phi$ and $\\mathcal{V}_{\\overline G}$, which\nare non-isomorphic in general (the group $\\mathcal{V}_{\\overline G}$ is a\nhomomorphic image of $\\mathcal{V}_\\Phi$).\n\n\\begin{proposition}\n\\label{pr:kerneln}\nLet $\\Phi$ be a covering biset. Denote by $K_n$ the subgroup of\nelements of $G$ acting trivially from the left on $\\Phi^{\\otimes n}$,\ni.e., the kernel of the wreath recursion associated with the biset\n$\\Phi^{\\otimes n}$. Then $K_n\\supseteq K_{n-1}$. If $\\bigcup_{n\\ge\n  0}K_n$ is equal to the kernel of the epimorphism\n$G\\longrightarrow\\overline{G}$, then the natural epimorphism\n$\\mathcal{V}_\\Phi\\longrightarrow\\mathcal{V}_{\\overline G}$ is an isomorphism.\n\\end{proposition}\n\n\\begin{proof}\nSuppose that an element $g$ of the kernel of $\\mathcal{V}_\\Phi\\longrightarrow\\mathcal{V}_{\\overline\n  G}$ is defined by a table $\\left(\\begin{array}{cccc}v_1 & v_2 &\n    \\ldots & v_n\\\\ g_1 & g_2 & \\ldots & g_n\\\\ u_1 & u_2 & \\ldots &\n    u_n\\end{array}\\right)$. Then the table $\\left(\\begin{array}{cccc}v_1 & v_2 &\n    \\ldots & v_n\\\\ \\overline{g_1} & \\overline{g_2} & \\ldots & \\overline{g_n}\\\\ u_1 & u_2 & \\ldots &\n    u_n\\end{array}\\right)$ represents the trivial element of\n$\\mathcal{V}_{\\overline G}$, where $g\\mapsto\\overline g$ is the epimorphism\n$G\\longrightarrow\\overline G$. But this means that $v_i=u_i$ and\n$\\overline{g_i}=1$ for all $i$. Consequently, there exists $k$ such\nthat $g_i\\in K_k$ for all $i$. It follows that after applying elementary\nsplittings $k$ times to all column of the table defining $g$, we will\nget a table defining the trivial element of $\\mathcal{V}_\\Phi$, which means\nthat $g=1$.\n\\end{proof}\n\n\n\\section{Simplicity of the commutator subgroup}\n\n\\subsection{Some general facts}\n\\label{ss:somegeneralfacts}\n\nLet $G$ be a group acting faithfully by homeomorphisms on an infinite\nHausdorff space $\\mathcal{X}$. For an open subset $U\\subset\\mathcal{X}$, denote\nby $G_{(U)}$ the subgroup of elements of $G$ acting trivially on\n$\\mathcal{X}\\setminus U$. \nDenote by $R_U$ the normal closure in $G$ of the derived subgroup\n$G_{(U)}{}'=[G_{(U)}, G_{(U)}]$.\n\nThe following simple lemma has appeared in many papers in different\nforms, see, for example \\cite[Lemma~5.3]{handbook:branch}, \n\\cite[Theorem~4.9]{matui:fullI}.\n\n\\begin{lemma}\n\\label{lem:basic}\nLet $N$ be a non-trivial normal subgroup of $G$. Then there exists a\nnon-empty open subset $U\\subset\\mathcal{X}$ such that $R_U\\le N$.\n\\end{lemma}\n\n\\begin{proof}\nIt is sufficient to prove that there exists an open subset $U$ such\nthat $G_{(U)}{}'\\le N$.\n\nLet $g\\in N\\setminus\\{1\\}$, and let $x\\in\\mathcal{X}$ be such that\n$g(x)\\ne x$. Then there exists an open subset $U$ such that $x\\in U$\nand $U\\cap gU=\\emptyset$. For example, find disjoint neighborhoods $U_x$\nand $U_{g(x)}$ of $x$ and $g(x)$, and set $U=U_x\\cap\ng^{-1}(U_{g(x)})$. \n\nLet $h_1, h_2\\in G_{(U)}$. Then $gh_1^{-1}g^{-1}$ acts trivially\noutside $gU$. Consequently, $[g^{-1}, h_1]=gh_1^{-1}g^{-1}h_1$ acts as $h_1$\non $U$, as $gh_1^{-1}g^{-1}$ on $gU$, and trivially outside \n$U\\cup gU$.  It follows that $[[g^{-1}, h_1], h_2]$ acts as $[h_1, h_2]$\non $U$ and trivially outside $U$, i.e., $[[g^{-1}, h_1], h_2]=[h_1,\nh_2]$. On the other hand, $[[g^{-1}, h_1], h_2]\\in N$, since $N$ is normal\nand $g\\in N$. It follows that $[h_1, h_2]\\in N$ for all $h_1, h_2\\in\nG_{(U)}$, i.e., that $G_{(U)}{}'\\le N$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lem:Ru}\nLet $U$ be an open subset of $\\mathcal{X}$ such that its $G$-orbit is\na basis of topology of $\\mathcal{X}$. Then every non-trivial normal subgroup of\n$G$ contains $R_U$.\n\\end{lemma}\n\n\\begin{proof}\nFor any $g\\in G$ and $U\\subset\\mathcal{X}$, we have\n$gG_{(U)}g^{-1}=G_{(gU)}$. Consequently,\n$gG_{(U)}{}'g^{-1}=G_{(gU)}{}'$, and $R_U$ is\nequal to the group generated by $\\bigcup_{g\\in G}G_{(gU)}{}'$. In\nparticular, $R_{gU}=R_U$ for all $g\\in G$.\n\nSince $\\{gU\\;:\\;g\\in G\\}$ is a basis of topology, for every open\nsubset $W\\subset\\mathcal{X}$ there exists $g\\in G$ such that $gU\\subseteq W$,\nhence $R_U=R_{gU}\\le R_W$. This implies, by Lemma~\\ref{lem:basic},\nthat $R_U$ is contained in every non-trivial normal subgroup of $G$.\n\\end{proof}\n\n\\begin{proposition}\n\\label{pr:RU}\nSuppose that $U$ is an open subset of $\\mathcal{X}$ such that its $R_U$-orbit\nis equal to its $G$-orbit and\nis a basis of topology of $\\mathcal{X}$. Then $R_U$ is simple and is contained\nin every non-trivial normal subgroup of $G$.\n\\end{proposition}\n\n\\begin{proof}\nThe group $G_{(U)}{}'$ is non-trivial, since\notherwise its normal closure $R_U$ is trivial, which contradicts the\nfact that the $R_U$-orbit of $U$ is a basis of topology. It follows that\nthere exists $g\\in G_{(U)}{}'$ moving a point $x\\in\\mathcal{X}$. We have then $x\\in U$\nand $g(x)\\in U$. There exists a neighborhood $W$ of $x$ such that\n$W\\cap gW=\\emptyset$. Then there exists an element $h\\in R_U$ such\nthat $hU\\subset W$, and there exists a non-trivial element $g'\\in\nG_{(hU)}{}'\\le G_{(U)}{}'$. We have $[g, g']\\ne 1$, since $g'$ and $gg'g^{-1}$\nhave disjoint supports. This shows that $G_{(U)}{}'$ is non-abelian,\ni.e., that $G_{(U)}{}''$ is non-trivial.\n\nThe subgroup $R_U$ is contained in every\nnon-trivial normal subgroup of $G$, by Lemma~\\ref{lem:Ru}.\nWe also have that the normal closure in $R_U$ of the group\n$(R_U)_{(U)}{}'$ is contained in every normal subgroup of $R_U$.\n\nNote that $G_{(U)}{}'$ is contained in $G_{(U)}\\cap R_U=(R_U)_{(U)}$,\nhence $G_{(U)}{}''\\le(R_U)_{(U)}{}'$. Conjugating by an element $g\\in\nG$ (and using that\n$R_U$ is normal in $G$), we get that $G_{(gU)}{}''\\le (R_U)_{(gU)}{}'$.\n\nIt follows that for any non-trivial subgroup $N\\unlhd R_U$ we have\n \\[N\\supseteq\\bigcup_{g\\in\n  R_U}(R_U)_{(gU)}{}'\\supseteq\\bigcup_{g\\in R_U}G_{(gU)}{}''=\\bigcup_{g\\in\n  G}G_{(gU)}{}''.\\]\nConsequently, $N$ contains the group generated by the set $\\bigcup_{g\\in\n  G}G_{(gU)}{}''$, which is normal in $G$ and non-trivial, hence\ncontains $R_U$. We have proved that every non-trivial normal subgroup of $R_U$\ncontains $R_U$, i.e., that $R_U$ is simple.\n\\end{proof}\n\n\\subsection{Simplicity of $\\mathcal{V}_G'$}\n\nLet $(G, X)$ be a self-similar group, and let $\\mathcal{V}_G$ be the corresponding\ngroup of homeomorphisms of the Cantor set $X^\\omega$.\n\nFix a linear ordering ``$<$'' of the elements of\n$X$. Extend it to the lexicographic ordering on $X^*$. Namely, if\n$x_1x_2\\ldots x_n$ and $y_1y_2\\ldots y_m$ are incomparable with\nrespect to the order $\\preceq$, then\n$x_1x_2\\ldots x_n<y_1y_2\\ldots y_m$ if and only if $x_i<y_i$, where\n$i$ is the smallest index such that $x_i\\ne y_i$. If a word $v$ is a\nbeginning of a word $w$, then $v\\le w$.\n\nSuppose that $\\left(\\begin{array}{cccc}v_1 & v_2 & \\ldots & v_n\\\\ u_1 &\nu_2 & \\ldots & u_n\\end{array}\\right)$ is a table defining an\nelement $g\\in\\mathcal{V}_X$, i.e., that $g=\\sum_{i=1}^nS_{u_i}S_{v_i}^*$.\nAssume that its first row is ordered in the increasing\nlexicographic order. If the permutation putting the second row into\nthe increasing lexicographic order is even, then we say that the table\nis even.\n\nNote that if $d=|X|$ is even, then every table has an even splitting. On the\nother hand, if $d$ is odd, then the set of elements defined by even\ntables is a subgroup of index 2 in $\\mathcal{V}_d$. The following is proved\nin~\\cite{hgthomp}.\n\n\\begin{theorem}\n\\label{th:vdprime}\nIf $d$ is even, then $\\mathcal{V}_d$ is simple. If $d$ is odd, then the\ncommutator subgroup $\\mathcal{V}_d'$ is the group of elements defined by even\ntables, and is a simple subgroup of index 2 in $\\mathcal{V}_d$.\n\\end{theorem}\n\nThe following theorem is a proved in~\\cite[Theorem~9.11]{nek:bim}.\n\n\\begin{theorem}\n\\label{th:vgabelian}\nAll proper quotients of $\\mathcal{V}_G$ are abelian.\n\\end{theorem}\n\nLet us describe the $\\mathcal{V}_d$-orbits of clopen subsets of $X^\\omega$.\n\n\\begin{proposition}\n\\label{pr:VXorbits}\nLet $d=|X|$, and let $U$ be a clopen subset of $X^\\omega$. Let us\ndecompose $U$ into a disjoint union $\\bigsqcup_{i=1}^n v_iX^\\omega$\nfor $v_i\\in X^*$. Then the residue of $n$ modulo $d-1$ does not depend\non the decomposition. Let us denote it $m(U)\\in\\mathbb{Z}\/(d-1)\\mathbb{Z}$. Let $U_1,\nU_2\\subset X^\\omega$ be non-empty clopen proper subsets. Then the\nfollowing conditions are equivalent.\n\\begin{enumerate}\n\\item $U_1$ and $U_2$ belong to one $\\mathcal{V}_X$-orbit.\n\\item $U_1$ and $U_2$ belong to one $\\mathcal{V}_X'$-orbit.\n\\item $m(U_1)=m(U_2)$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nLet $U=\\bigsqcup_{i=1}^n v_iX^\\omega$ be a decomposition of $U$. We can\n\\emph{split} it by replacing one set $v_iX^\\omega$ by the collection\nof $d$ sets $v_ixX^\\omega$, for $x\\in X$. This way we increase the\nnumber of sets in the decomposition by $d-1$. It is easy to see that\nfor any two decompositions of $U$ into cylindrical sets \nthere exist sequences of successive splittings of each of\nthe decompositions leading to the same decomposition. This implies\nthat $m(U)$ is well defined.\n\nIt is obvious that $m(U)$ is preserved under the action of\n$\\mathcal{V}_X$. Suppose that $m(U_1)=m(U_2)$ for non-empty proper clopen\nsubsets of $X^\\omega$. Then there exist decompositions\n$U_1=\\bigsqcup_{i=1}^nv_iX^\\omega$ and $U_2=\\bigsqcup_{i=1}^n\nu_iX^\\omega$ of the sets $U_i$ into equal number of cylindrical\nsubsets. The sets $\\{v_i\\}_{i=1}^n$ and $\\{u_i\\}_{i=1}^n$ are\nincomplete antichains in $X^*$, hence (see Lemma~\\ref{lem:incomplete})\nthere exists $g\\in \\mathcal{V}_X$ such that\n$g(v_iX^\\omega)=u_iX^\\omega$ for all $i$. If $g\\notin\\mathcal{V}_X'$, then\nwe can compose it with an element $h\\in\\mathcal{V}_X\\setminus\\mathcal{V}_X'$ acting\ntrivially on $\\bigsqcup_{i=1}^nv_iX^\\omega$, and get an element\n$gh\\in\\mathcal{V}_X'$ such that $gh(v_iX^\\omega)=u_iX^\\omega$ for all $i$. It\nfollows that $U_1$ and $U_2$ belong to one $\\mathcal{V}_X'$-orbit and to one\n$\\mathcal{V}_X$-orbit.\n\\end{proof}\n\nIt easily follows from the definitions that $\\mathcal{V}_X$-orbits of clopen\nsubsets of $X^\\omega$ coincide with their $\\mathcal{V}_G$-orbits for any\nself-similar group $(G, X)$.\n\n\\begin{theorem}\n\\label{th:commutatorsimple}\nThe commutator subgroup of $\\mathcal{V}_G$ is simple.\n\\end{theorem}\n\n\\begin{proof}\nLet $U=xX^\\omega$ for some $x\\in X$. Then the $\\mathcal{V}_G$-orbit of $U$\ncoincides with its $\\mathcal{V}_X$-orbit and its $\\mathcal{V}_X'$-orbit, and is a basis\nof the topology on $X^\\omega$, see Proposition~\\ref{pr:VXorbits}. The\ngroup $\\mathcal{V}_X'$ is contained in $R_U$, since the normal closure of\n$(\\mathcal{V}_X)_{(U)}{}'$ in $\\mathcal{V}_X$ is equal to $\\mathcal{V}_X'$, by\nTheorem~\\ref{th:vdprime}. It follows that the $R_U$-orbit of $U$ is\nequal to its $G$-orbit, and is a basis of the topology.\n\nConsequently, by Proposition~\\ref{pr:RU}, $R_U$ is simple and is\ncontained in every non-trivial normal subgroup of $\\mathcal{V}_G$. On the other hand, by\nTheorem~\\ref{th:vgabelian}, every non-trivial normal subgroup of\n$\\mathcal{V}_G$ contains the commutator subgroup $\\mathcal{V}_G{}'$. It follows that\n$\\mathcal{V}_G'=R_U$, and $\\mathcal{V}_G'$ is simple.\n\\end{proof}\n\nAbelianization of $\\mathcal{V}_G$ is described in~\\cite[Theorem~9.14]{nek:bim}.\n\n\\begin{theorem}\n\\label{th:abelquotient}\nLet $(G, X)$ be a self-similar group. Let $\\pi:G\\longrightarrow G\/G'$ be the\nabelianization epimorphism.\n\nSuppose that $d$ is even. Then $\\mathcal{V}_G\/\\mathcal{V}_G'$ is isomorphic to the\nquotient of $G\/G'$ by the relations\n$\\pi(g)=\\sum_{x\\in X}\\pi(g|_x)$ for all $g\\in G$.\n\nSuppose that $d$ is odd. Then $\\mathcal{V}_G\/\\mathcal{V}_G'$ is isomorphic to\n$\\mathbb{Z}\/2\\mathbb{Z}\\oplus G\/G'$ modulo the relations\n$\\pi(g)=\\mathop{\\mathrm{sign}}(g)\\oplus\\sum_{x\\in X}\\pi(g|_x)$ for\nall $g\\in G$; where $\\mathop{\\mathrm{sign}}(g)=0$ if $g$ defines an even permutation on\nthe first level $X$ of $X^*$, and $\\mathop{\\mathrm{sign}}(g)=1$ otherwise.\n\\end{theorem}\n\nProof of the following lemma follows directly from the multiplication\nrule~\\eqref{eq:multiplicationwreath}.\n\n\\begin{lemma} \n\\label{lem:transfer}\nLet $(G, X)$ be a self-similar group, and let\n$\\pi:G\\longrightarrow G\/G'$ be the canonical homomorphism.\nThen $\\sigma:\\pi(g)\\mapsto\\sum_{x\\in X}\\pi(g|_x)$ is a\nwell defined endomorphism of $G\/G'$.\n\\end{lemma}\n\nFor odd $d$ denote by $\\sigma_1$ the endomorphism $(t, g)\\mapsto\n(t+\\mathop{\\mathrm{sign}}(g), \\sigma(g))$ of $\\mathbb{Z}\/2\\mathbb{Z}\\oplus G\/G'$, where\n$\\mathop{\\mathrm{sign}}:G\/G'\\longrightarrow\\mathbb{Z}\/2\\mathbb{Z}$ gives the parity of the action\nof any preimage of $g$ in $G$ on the first level $X$ of the tree $X^*$.\n\nThen Theorem~\\ref{th:abelquotient} can be reformulated as follows. (We\ndenote here the identical automorphism of a group by $1$.)\n\n\\begin{theorem}\n\\label{th:abelquotient2}\nIf $d$ is even, then $\\mathcal{V}_G\/\\mathcal{V}_G'$ is isomorphic to the\nquotient of $G\/G'$ by the range of the endomorphism $1-\\sigma$.\n\nIf $d$ is odd, then $\\mathcal{V}_G\/\\mathcal{V}_G'$ is isomorphic to the\nquotient of $\\mathbb{Z}\/2\\mathbb{Z}\\oplus G\/G'$ by the range of the endomorphism $1-\\sigma_1$.\n\\end{theorem}\n\n\\begin{example}\nConsider the double self-covering $f(x)=2x$ of the circle\n$\\mathbb{R}\/\\mathbb{Z}$. Then $\\img{f}$ is generated by the adding machine\n$a=S_1S_0^*+S_0aS_1^*$.\nIt follows that $\\mathcal{V}_f\/\\mathcal{V}_f'$ is the quotient of $\\mathbb{Z}=\\img{f}\/\\img{f}'$ by the range of\n$1-\\sigma$, where $\\sigma(a)=0+a=a$ is the identity\nisomorphism. Hence, $1-\\sigma=0$, and $\\mathcal{V}_f\/\\mathcal{V}_f'\\cong\\mathbb{Z}$.\n\\end{example}\n\n\\begin{example}\nLet $G=\\mathfrak{K}_{x_1x_2\\ldots x_{n-1}}$ be the iterated monodromy\ngroup of a quadratic polynomial, as in Example~\\ref{ex:kneadingv}. It\nis proved in~\\cite[Proposition~3.3]{bartnek:mand}\nthat $G\/G'$ is the free abelian group $\\mathbb{Z}^n$ freely generated by\nthe images of the generators $a_i$ of $G$. The recursive\nrelations defining $\\mathfrak{K}_{x_1x_2\\ldots x_{n-1}}$ show that\n$\\mathcal{V}_G\/\\mathcal{V}_G'$ is $\\mathbb{Z}^n=\\langle\\pi(a_0), \\ldots, \\pi(a_{n-1})\\rangle$\nmodulo the relations $\\pi(a_0)=\\pi(a_1)$, and $\\pi(a_i)=\\pi(a_{i-1})$\nfor all $i=1, 2, \\ldots, n-1$. Consequently, $\\mathcal{V}_G\/\\mathcal{V}_G'$ is\nisomorphic to $\\mathbb{Z}$.\n\\end{example}\n\n\n\\begin{example}\nLet $G$ be the Grigorchuk group, see Example~\\ref{ex:GrigorchukG}.\nIt is generated by \n\\[a=S_1S_0^*+S_0S_1^*, b=S_0aS_0^*+S_1cS_1^*,\nc=S_0aS_0^*+S_1dS_1^*, d=S_0S_0^*+S_1bS_1^*.\\]\n\nIt follows that $\\mathcal{V}_G\/\\mathcal{V}_G'$ is the group $G\/G'$ modulo the relations\n\\[\\pi(a)=0, \\pi(b)=\\pi(a)+\\pi(c), \\pi(c)=\\pi(a)+\\pi(d),\n\\pi(d)=\\pi(c),\\]\nwhich are equivalent to $\\pi(a)=0, \\pi(b)=\\pi(c)=\\pi(d)$. It is easy\nto check that $b=cd$ and $b^2=1$ in $G$, which implies\n\\[\\pi(c)=\\pi(d)=\\pi(b)=\\pi(c)+\\pi(d)=\\pi(b)+\\pi(b)=\\pi(b^2)=0.\\]\nConsequently, $\\mathcal{V}_G\/\\mathcal{V}_G'$ is trivial, hence $\\mathcal{V}_G$ is\nsimple. Simplicity of $\\mathcal{V}_G$ was proved in~\\cite{roever}.\n\\end{example}\n\n\\section{Expanding maps and finite presentation}\n\n\\subsection{Contracting self-similar groups and expanding maps}\n\n\\begin{defi}\n\\label{def:nucleus}\nLet $(G, X)$ be a self-similar group. We say that it is\n\\emph{contracting} if there exists a finite set $\\mathcal{N}\\subset G$ such\nthat for every $g\\in G$ there exists $n$ such that $g|_v\\in\\mathcal{N}$ for\nall $v\\in X^*$, $|v|\\ge n$.\n\\end{defi}\n\nMore generally, let $\\Phi$ be a covering $G$-biset, and let $X$ be a\nright orbit transversal. Define then $g|_v$ for $g\\in G$ and $v\\in\nX^n\\subset\\Phi^{\\otimes n}$ as the unique element of $G$ such that\n$g\\cdot v=u\\cdot g|_v$ for some (also unique) $u\\in X^n$. We say that\n$\\Phi$ is contracting (or \\emph{hyperbolic}) if there exists a finite\nset $\\mathcal{N}$ such that for every $g\\in G$ there exists $n$ such that\n$g|_v\\in\\mathcal{N}$ for all $v\\in X^*$ such that $|v|\\ge n$. One can show\n(see~\\cite[Corollary~2.11.7]{nek:book}) that this property depends only on $\\Phi$ (but the set\n$\\mathcal{N}$ will depend on $X$).\n\nThe smallest set $\\mathcal{N}$ satisfying the conditions of\nDefinition~\\ref{def:nucleus} is called the \\emph{nucleus} of the\nself-similar action.\n\n\\begin{defi}\n\\label{def:expandingmap}\nLet $f:\\mathcal{M}_1\\longrightarrow\\mathcal{M}$ be a partial self-covering such that $\\mathcal{M}$ is\ncompact.\nThe covering is called\n\\emph{expanding} if there exist $L>1$, $\\epsilon>0$, a positive\ninteger $n$, and a metric $|x-y|$ on $\\mathcal{M}$ such\nthat \\[|f^n(x)-f^n(y)|\\ge L|x-y|\\]\nfor all $x, y\\in\\mathcal{M}_n$ such that $|x-y|<\\epsilon$.\n\\end{defi}\n\nFor example, if $\\mathcal{M}$ is a connected Riemann manifold, and\n$\\|Df^n(\\xi)\\|\\ge CL^n\\|\\xi\\|$ for all $n\\ge 1$ and everly tangent vector\n$\\xi$, then $f$ is expanding.\n\nThe following proposition is proved in~\\cite[Theorem~5.5.3]{nek:book}\nfor the case when $\\mathcal{M}$ is a complete length metric space with finitely\ngenerated fundamental group. We will repeat here the proof in a more\ngeneral situation (but avoiding orbispaces, which was one of the technical\nissues in~\\cite{nek:book}).\n\n\\begin{proposition}\n\\label{pr:expcontracting}\nLet $f:\\mathcal{M}_1\\longrightarrow\\mathcal{M}$ be an expanding partial self-covering of a compact path\nconnected space. Then $\\img{f}$ is a contracting self-similar group\n(with respect to any standard action).\n\\end{proposition}\n\n\\begin{proof}\nLet $\\{\\ell_i\\}_{i=1, \\ldots, d}$ be paths connecting the basepoint\n$t\\in\\mathcal{M}$ to the preimages $z_i\\in f^{-1}(t)$. We consider the standard\naction of $\\img{f}$ on $X^*$ defined by these connecting paths, where\n$X=\\{1, \\ldots, d\\}$.\n\nIt follows from Definition~\\ref{def:expandingmap} that\nthere exist $\\epsilon>0, L>1, C>1$ such that for any subset $A\\subset\\mathcal{M}$ of\ndiameter less than $\\epsilon$ and every $n\\ge 1$, the set $f^{-n}(A)$ is a\ndisjoint union of $d^n$ sets $A_i$ such that $f^n:A_i\\longrightarrow A$ are\nhomeomorphisms, and diameters of $A_i$ are less than $CL^{-n}$.\n\nLet $g\\in\\img{f}$ be defined by a loop $\\gamma$. Since we can\nrepresent $\\gamma$ as a union of sub-paths of diameter less than\n$\\epsilon$, there exists $C_\\gamma>1$ such that\ndiameter of any lift of $\\gamma$ by $f^n$ is less than $C_\\gamma L^{-n}$.\n\nBy Lemma~\\ref{lem:ssimilarity}, section $g|_{i_1i_2\\ldots i_n}$ is\ndefined by the loop $\\ell_{j_1j_2\\ldots j_n}^{-1}\\gamma_{i_1i_2\\ldots\n  i_n}\\ell_{i_1i_2\\ldots i_n}$, where $\\gamma_{i_1i_2\\ldots i_n}$ is a\nlift of $\\gamma$ by $f^n$, and  $\\ell_{i_1i_2\\ldots i_n}$,\n$\\ell_{j_1j_2\\ldots j_n}$ are paths inside the tree $\\Gamma_t$ formed\nby lifts of the connecting paths $\\ell_i$. Since diameters of lifts of\npaths by $f^n$ exponentially decrease with $n$, there exists $n_0$\n(depending only on the connecting paths $\\ell_i$) such that if $n_0\\le\nn<m$, then for all\nsequences $i_1i_2\\ldots i_m\\in X^*$, the\npath $\\ell_{i_1i_2\\ldots i_m}$ is a continuation of the path\n$\\ell_{i_1i_2\\ldots i_n}$ by a path of diameter less than\n$\\epsilon\/6$.\n\nThere exists $n_1\\ge n_0$ (depending on $\\gamma$) \nsuch that if $n\\ge n_1$, then all lifts of $\\gamma$\nby $f^n$ have diameters less than $\\epsilon\/6$. Then for every $n\\ge\nn_1$, the section $g|_{i_1i_2\\ldots i_n}$ is defined by a loop of the\nform $\\beta=\\ell_{j_1j_2\\ldots j_{n_0}}^{-1}\\alpha\\ell_{i_1i_2\\ldots i_{n_0}}$,\nwhere $\\alpha$ is a path of diameter less than\n$\\epsilon\/2$. If $\\beta'=\\ell_{j_1j_2\\ldots j_{n_0}}^{-1}\n\\alpha'\\ell_{i_1i_2\\ldots i_{n_0}}$ is another path such\nthat $\\alpha'$ has diameter less than $\\epsilon\/2$, then\n\\[\\beta'\\beta^{-1}=\\ell_{j_1j_2\\ldots\n  j_{n_0}}^{-1}\\alpha'\\alpha^{-1}\\ell_{j_1j_2\\ldots j_{n_0}},\\]\nwhere $\\alpha'\\alpha^{-1}$ is a loop of diameter less than\n$\\epsilon$. By the choice of $\\epsilon$, all lifts of $\\alpha'\\alpha^{-1}$\nby iterations of $f$ are loops, which implies that $\\beta$ and\n$\\beta'$ define equal elements of $\\img{f}$. Consequently,\n$g|_{i_1i_2\\ldots i_n}$ is uniquely determined by the pair\n$(i_1i_2\\ldots i_{n_0}, j_1j_2\\ldots j_{n_0})$. Since $n_0$ does not\ndepend on $g$, it follows that $\\img{f}$ is contracting.\n\\end{proof}\n\n\\subsection{General definition of $\\mathcal{V}_f$ for expanding maps}\n\\label{ss:generaldefinition}\n\nThe group $\\mathcal{V}_f$ can be defined for an expanding self-covering\n$f:\\mathcal{M}\\longrightarrow\\mathcal{M}$, even if $\\mathcal{M}$ is not path connected.\n\nLet $f:\\mathcal{M}\\longrightarrow\\mathcal{M}$ be an expanding self-covering of a compact metric\nspace $\\mathcal{M}$, such that for every $t\\in\\mathcal{M}$ the set $\\bigcup_{n\\ge\n  0}f^{-n}(t)$ is dense in $\\mathcal{M}$. This condition is always satisfied\nfor a path-connected space $\\mathcal{M}$.\n\nLet $\\delta>0$ be such that for any two points $t_1, t_2\\in\\mathcal{M}$ such\nthat $t_1\\ne t_2$ and $f(t_1)=f(t_2)$ we have $|t_1-t_2|>\\delta$. It\nis easy to prove that such $\\delta$ exists for any self-covering\n$f:\\mathcal{M}\\longrightarrow\\mathcal{M}$ of a compact metric space. It follows from the definition\nof an expanding self-covering, that there exists $\\epsilon>0$ such\nthat for any two points $z_1, z_2\\in\\mathcal{M}$ and for any $n\\ge 1$ there\nexists an isomorphism $S_{z_1, z_2}:T_{z_1}\\longrightarrow T_{z_2}$ of the trees\nof preimages such that $|S_{z_1, z_2}(v)-v|<\\delta\/2$ for all $v\\in\nT_{z_1}$. Moreover, it is easy to prove (by induction on the level number)\nthat the isomorphism $S_{z_1, z_2}$ is unique.\n\nFix a basepoint $t\\in\\mathcal{M}$, and define $\\mathcal{V}_f$ as the group of\nhomeomorphisms of $\\partial T_t$ piecewise equal to the isomorphisms\n$S_{z_1, z_2}:\\partial T_{z_1}\\longrightarrow\\partial T_{z_2}$ for $z_1, z_2\\in\nT_t$.\n\nNote that if $\\gamma:[0, 1]\\longrightarrow\\mathcal{M}$ is a path in $\\mathcal{M}$, then there\nexists $n$ such that all lifts of $\\gamma$ by $f^m$ for $m\\ge n$ have\ndiameter less than $\\delta\/2$. This implies that if $\\mathcal{M}$ is path\nconnected, then our original definition of $\\mathcal{V}_f$ agrees with the\ngiven definition for expanding maps.\n\n\\begin{example}\nConsider the one-sided shift $\\mathsf{s}:X^\\omega\\longrightarrow X^\\omega$ \n\\[\\mathsf{s}(x_1x_2\\ldots)=x_2x_3\\ldots.\\] Consider the metric\n$|w_1-w_2|=2^{-n}$, where $n$ is the smallest index for which $x_n\\ne\ny_n$, where $w_1=x_1x_2\\ldots$ and $w_2=y_1y_2\\ldots$.\n\nThen $|\\mathsf{s}(w_1)-\\mathsf{s}(w_2)|=2|w_1-w_2|$ whenever\n$|w_1-w_2|\\le 1\/2$. Consequently, $\\mathsf{s}$ is expanding. \n\nFor any\n$w\\in X^\\omega$ the set $\\mathsf{s}^{-n}(w)$ is equal to the set of\nsequences of the form $vw$, where $v\\in X^n$. Hence, we can identify\nthe $n$th level of the tree $T_w$ with the set $X^n$ by the map\n$vw\\mapsto v$. Note that the tree $T_w$ after this identification\nbecomes the left Cayley graph of the monoid $X^*$: two vertices are\nconnected by an edge if and only if they are of the form $v, xv$ for\n$v\\in X^*$, $x\\in X$. In particular, the boundary $\\partial T_w$ is\nnaturally identified with the space $X^{-\\omega}$ of left-infinite\nsequences $\\ldots x_2x_1$.\n\nIt follows directly from the definitions that\nthe maps $S_{wv_1, wv_2}:T_{wv_1}\\longrightarrow T_{wv_2}$ act by the rule\n\\[S_{wv_1, wv_2}(uv_1)=uv_2.\\]\nIt follows that the homeomorphism $\\ldots x_2x_1\\mapsto x_1x_2\\ldots$\nof $X^{-\\omega}=\\partial T_w$ with $X^\\omega$ conjugates the action of\n$\\mathcal{V}_{\\mathsf{s}}$ with the Higman-Thompson group $\\mathcal{V}_X$.\n\\end{example}\n\n\\begin{example}\nIf $\\mathcal{M}$ is not connected, then a covering $f:\\mathcal{M}\\longrightarrow\\mathcal{M}$ needs not to be\nof constant degree. For example, $f:\\mathcal{M}\\longrightarrow\\mathcal{M}$ can be a one-sided shift\nof finite type. The corresponding group $\\mathcal{V}_f$ is the topological\nfull groups of a shift of finite type (the dual of $f$). These groups were\nstudied in~\\cite{matui:fullonesided}.\n\\end{example}\n\n\\subsection{Abelianization of $\\mathcal{V}_f$ in expanding case}\n\nSelf-similar contracting groups acting faithfully on $X^*$ are\ntypically infinitely presented. \nOn the other hand, for every contracting group $G$ there exists a\nfinitely presented group $\\tilde G$ and a hyperbolic covering $\\tilde G$-biset\n$\\Phi$ such that the faithful quotient of $\\tilde G$ is $G$. More\nprecisely, we have the following description of $\\tilde G$, given\nin~\\cite[Section~2.13.2.]{nek:book}.\n\n\\begin{proposition}\n\\label{pr:lengththree}\nLet $(G, X)$ be a contracting group. Suppose that the nucleus $\\mathcal{N}$ \ngenerates $G$.\n\nLet $\\tilde G$ be the group given by the presentation $\\langle\n\\mathcal{N}\\;|\\;R\\rangle$, where $R$ is the set of all relations\n$g_1g_2g_3=1$ of length at most 3 that hold for elements of $\\mathcal{N}$ in\n$G$. Let $\\Phi$ be the $\\tilde G$-biset of pairs $x\\cdot g$, for $x\\in\nX$ and $g\\in\\tilde G$ with the actions given by the usual rules:\n\\[(x\\cdot g)\\cdot h=x\\cdot (gh),\\qquad h\\cdot (x\\cdot g)=h(x)\\cdot\n(h|_xg),\\]\nwhere $g\\in\\tilde G$, $h\\in\\mathcal{N}$, $x\\in X$; and $h(x)\\in X$, $h|_x\\in\\mathcal{N}$ are\ndefined as in $G$. Then $\\Phi$ is contracting.\n\\end{proposition}\n\nNote that for any contracting group $G$, the group generated by the\nnucleus $\\mathcal{N}$ is self-similar contracting, and\n$\\mathcal{V}_{\\langle\\mathcal{N}\\rangle}=\\mathcal{V}_G$.\n\nThe following is proved in~\\cite[Proposition~2.13.2]{nek:book}.\n\n\\begin{proposition}\n\\label{pr:kernelcontracting}\nLet $\\Phi$ be a contracting $G$-biset. Let $\\rho:G\\longrightarrow\\overline G$ be\nthe canonical epimorphism onto the faithful quotient of $G$. If\n$\\rho(g)\\ne 1$ for every non-trivial element of the nucleus of $G$\n(defined using some right orbit transversal $X$),\nthen the kernel of $\\rho$ is equal to the union of the kernels $K_n$ \nof the left actions of $G$ on $\\Phi^{\\otimes n}$.\n\\end{proposition}\n\nLet $\\Phi$ be a $G$-biset, and let $d$ be the number of\norbits of the right action of $G$ on $\\Phi$. Choose a right orbit\ntransversal $X\\subset\\Phi$, and define, for $g\\in G$ and $x\\in X$, the\nsection $g|_x$ as the unique element of $G$ such that $g\\cdot x=y\\cdot\ng|_x$ for $y\\in X$. Let $\\pi:G\\longrightarrow G\/G'$ be the abelianization\nepimorphism.\n\nBy the same arguments as in Lemma~\\ref{lem:transfer}, \nthe map $\\sigma:\\pi(g)\\mapsto\\sum_{x\\in X}\\pi(g|_x)$ is a well defined\nendomorphism of $G\/G'$. It is also checked directly that it does not\ndepend on the choice of the right orbit transversal $X$.\nIf $d$ is odd, then define homomorphism\n$\\mathop{\\mathrm{sign}}:G\/G'\\longrightarrow\\mathbb{Z}\/2\\mathbb{Z}$ as in\nTheorem~\\ref{th:abelquotient}.\n\nThe following is a direct corollary of\nPropositions~\\ref{pr:kernelcontracting},~\\ref{pr:kerneln}, and\nTheorem~\\ref{th:abelquotient2}. \n\n\\begin{corollary}\n\\label{cor:abelquotcontracting}\nLet $\\Phi$ be a $G$-biset satisfying the conditions of\nProposition~\\ref{pr:kernelcontracting}.\n\nIf $d$ is even, then $\\mathcal{V}_\\Phi\/\\mathcal{V}_\\Phi'$ is isomorphic to the quotient\nof $G\/G'$ by the range of the homomorphism $1-\\sigma$. If $d$ is odd,\nthen $\\mathcal{V}_\\Phi\/\\mathcal{V}_\\Phi'$ is isomorphic to the quotient of $\\mathbb{Z}\/2\\mathbb{Z}\\oplus\nG\/G'$ by the range of the endomorphism $1-\\sigma_1$, where\n$\\sigma_1(t, g)=(t+\\mathop{\\mathrm{sign}}(g), \\sigma(g))$.\n\\end{corollary}\n\n\n\\begin{proposition}\n\\label{pr:expandingpi1}\nSuppose that $f:\\mathcal{M}_1\\longrightarrow\\mathcal{M}$ is expanding, $\\mathcal{M}$ is path-connected and\nsemi-locally simply connected. Then the $\\pi_1(\\mathcal{M})$-biset associated with $f$\nis contracting.\n\nIf $g\\in\\pi_1(\\mathcal{M})$ has trivial image in $\\img{f}$, then there exists\n$n$ such that $g$ acts trivially from the left on $\\Phi_f^{\\otimes\n  n}$.\n\\end{proposition}\n\nThe first paragraph of the proposition is proved in the same way as \nProposition~\\ref{pr:expcontracting}. The second paragraph follows\ndirectly from exponential decreasing of diameters of lifts of paths by\niterations of $f$ and the condition that $\\mathcal{M}$ is semi-locally simply\nconnected.\n\n\\begin{corollary}\nSuppose that $f:\\mathcal{M}_1\\longrightarrow\\mathcal{M}$ satisfies the conditions of\nProposition~\\ref{pr:expandingpi1}, and let $\\Phi$ be the\n$\\pi_1(\\mathcal{M})$-biset associated with $f$. Then $\\mathcal{V}_\\Phi$ is isomorphic to\n$\\mathcal{V}_f=\\mathcal{V}_{\\img{f}}$.\n\\end{corollary}\n\nLet $f:\\mathcal{M}_1\\longrightarrow\\mathcal{M}$ be a partial self-covering satisfying the conditions of\nProposition~\\ref{pr:expandingpi1}. Let $\\iota:\\mathcal{M}_1\\longrightarrow\\mathcal{M}$ be the\nidentical embedding. \n\nThe group\n$\\pi_1(\\mathcal{M})\/\\pi_1(\\mathcal{M})'$ is naturally isomorphic to the first homology\ngroup $H_1(\\mathcal{M})$. The map $\\sigma:H_1(\\mathcal{M})\\longrightarrow H_1(\\mathcal{M})$ from\nCorollary~\\ref{cor:abelquotcontracting} is equal to $\\iota_*\\circ\nf^!$, where $f^!:H_1(\\mathcal{M})\\longrightarrow H_1(\\mathcal{M}_1)$ is the map (called the\n\\emph{transfer map}) given by the condition that image of a chain $c$\nis equal to its full preimage $f^{-1}(c)$.\n\nSuppose that $c\\in H_1(\\mathcal{M})$ is\ndefined by a loop $\\gamma$. Then parity of the monodromy action of\n$\\gamma$ on fibers of $f$ is well defined and generates a homomorphism\nfrom $H_1(\\mathcal{M})$ to $\\mathbb{Z}\/2\\mathbb{Z}$. Let us denote it by $\\mathop{\\mathrm{sign}}:H_1(\\mathcal{M})\\longrightarrow\\mathbb{Z}\/2\\mathbb{Z}$.\n\nThen the following description of $\\mathcal{V}_f\/\\mathcal{V}_f'$ \nfollows directly from Corollary~\\ref{cor:abelquotcontracting}.\n\n\\begin{proposition}\n\\label{pr:homology}\nSuppose that $f:\\mathcal{M}_1\\longrightarrow\\mathcal{M}$ is expanding, $\\mathcal{M}$ is path-connected and\nsemi-locally simply connected. \n\nIf $\\deg f$ is even, then $\\mathcal{V}_f\/\\mathcal{V}_f'$ is isomorphic to the quotient\nof $H_1(\\mathcal{M})$ by the range of the endomorphism $1-\\iota_*\\circ f^!$.\n\nIf $\\deg f$ is odd, then $\\mathcal{V}_f\/\\mathcal{V}_f'$ is isomorphic to the quotient of\n$\\mathbb{Z}\/2\\mathbb{Z}\\oplus H_1(\\mathcal{M})$ by the range of the endomorphism $1-\\sigma_1$,\nwhere $\\sigma_1(t, c)=(t+\\mathop{\\mathrm{sign}}(c), \\iota_*\\circ f^!(c))$.\n\\end{proposition}\n\n\n\\subsection{Example: Hyperbolic rational functions}\n\nLet $f$ be a complex rational function, and let $C_f$ be the set of\ncritical points of $f:\\widehat\\mathbb{C}\\longrightarrow\\widehat\\mathbb{C}$. The\n\\emph{post-critical set} of $f$ is the union\n$P_f=\\bigcup_{n\\ge 1}f^n(C_f)$ of forward orbits of critical values.\nSuppose that $P_f$ is finite (we say\nthen that $f$ is \\emph{post-critically finite}). \n\nLet us additionally suppose that every cycle of $f:P_f\\longrightarrow P_f$ contains a critical\npoint. Then $f$ is \\emph{hyperbolic}, i.e., is expanding on a\nneighborhood of its Julia set, see~\\cite[Section~19]{milnor}.\n\nOne can find disjoint open topological discs around points of\n$P_f$ such that if $\\mathcal{M}$ is the complement of the union of these discs\nin the Riemann sphere, then $\\mathcal{M}$ contains the Julia set of $f$,\n$\\mathcal{M}_1=f^{-1}(\\mathcal{M})\\subset\\mathcal{M}$, and there exists a\nmetric on $\\mathcal{M}$ such that $f:\\mathcal{M}_1\\longrightarrow\\mathcal{M}$ is strictly expanding. For\ninstance, one can take discs bounded by the equipotential lines of the\nbasins of attraction, see~\\cite[Section~9]{milnor}.\n\nThen $H_1(\\mathcal{M})$ is isomorphic to the quotient of the free group\n$\\mathbb{Z}^{|P_f|}$ generated by elements $a_z$, $z\\in P_f$,\ncorresponding to the boundaries of the discs, modulo the relation\n$\\sum_{z\\in P_f}a_z=0$. It is easy to see that the map\n$\\sigma=\\iota_*\\circ f^!$ acts by the rule \n\\[\\sigma(a_z)=\\sum_{y\\in f^{-1}(z)\\cap P_f}a_y.\\]\n\nSuppose now that $\\deg f$ is odd. We say that $z$ is a \\emph{critical\nvalue mod 2} if $|f^{-1}(z)|$ is even. Note that if $z$ is a\ncritical value mod 2, then it is a critical value, since then\n$|f^{-1}(z)|\\ne\\deg f$. In particular, all critical values mod 2\nbelong to $P_f$. It is also easy to see that $z$ is a critical value\nmod 2 if and only if the monodromy action of a small simple loop\naround $z$ is an odd permutation. Namely, lengths of cycles of the\nmonodromy action are equal local degrees of $f$ in the preimages of\n$z$. The sum of local degrees is equal to $\\deg f$, i.e., is odd,\nhence the number of odd local degrees is odd.\nParity of the monodromy action is equal\nto parity of the number of cycles of even length, which is equal to\nparity of $|f^{-1}(z)|$ minus the number of odd local degrees,\nwhich is equal to parity of $|f^{-1}(z)|+1$.\n\n\\begin{proposition}\n\\label{pr:hyperbolicrational}\nLet $f$ be a hyperbolic post-critically finite rational function. Let\n$k$ be the number of attracting cycles of $f$. Let $l$ be the greatest\ncommon divisor of their lengths.\n\nIf $\\deg f$ is even, then\n$\\mathcal{V}_f\/\\mathcal{V}_f'$ is isomorphic to $\\mathbb{Z}^{k-1}\\oplus\\mathbb{Z}\/l\\mathbb{Z}$.\n\nIf $\\deg f$ is odd, and there exists an attracting cycle $C$ such that\nthe number of critical values mod 2 whose forward $f$-orbits are\nattracted to $C$ is odd, then $\\mathcal{V}_f\/\\mathcal{V}_f'$ is also isomorphic to\n$\\mathbb{Z}^{k-1}\\oplus\\mathbb{Z}\/l\\mathbb{Z}$. Otherwise, $\\mathcal{V}_f\/\\mathcal{V}_f'\\cong\\mathbb{Z}\/2\\mathbb{Z}\\oplus\\mathbb{Z}^{k-1}\\oplus\\mathbb{Z}\/l\\mathbb{Z}$.\n\\end{proposition}\n\nNote that $\\mathbb{Z}^{k-1}\\oplus\\mathbb{Z}\/l\\mathbb{Z}$ coincides with the $K_1$-group of the\n$C^*$-algebraic analog of $\\mathcal{V}_f$, see~\\cite{nek:cpalg}. Its\n$K_0$-group $\\mathbb{Z}\/(d-1)\\mathbb{Z}$ has also appeared in our paper, see Proposition~\\ref{pr:VXorbits}.\n\n\\begin{proof}\nEvery attracting cycle of $f$ is superattracting (i.e., contains\ncritical points of $f$), hence it belongs to the post-critical set $P_f$.\n\nSuppose at first that $\\deg f$ is even.\nIf $z\\in P_f$ does not belong to a cycle of $f:P_f\\longrightarrow P_f$, then\nthere exists $n$ such that $f^{-n}(z)\\cap P_f=\\emptyset$ and hence\n$\\sigma^n(a_z)=0$. It follows that the images of such elements $a_z$\nunder the epimorphism $\\pi:H_1(\\mathcal{M})\\longrightarrow\\mathcal{V}_f\/\\mathcal{V}_f'$ are equal to zero.\n\nIf $C$ is a cycle of $f:P_f\\longrightarrow P_f$, then the images of $a_z$ in\n$\\mathcal{V}_f\/\\mathcal{V}_f'$, for $z\\in C$, satisfy the relations $\\pi(a_z)=\\pi(a_{f(z)})$, since we\nhave $\\sigma(a_{f(z)})=a_z$. It follows that $\\mathcal{V}_f\/\\mathcal{V}_f'$ is the\nquotient of $H_1(\\mathcal{M})$ by the relations making elements corresponding\nto the points of each cycle of $f:P_f\\longrightarrow P_f$ equal, and making equal\nto zero all elements corresponding to elements of $P_f$ not belonging\nto cycles. It follows that $\\mathcal{V}_f\/\\mathcal{V}_f'$ is the quotient of the free\nabelian group $\\mathbb{Z}^k=\\langle e_1, e_2, \\ldots, e_k\\rangle$ by the\nrelation $l_1e_1+l_2e_2+\\cdots+l_ke_k=0$, where $l_i$ are the lengths\nof the corresponding cycles of $f:P_f\\longrightarrow P_f$. Consequently,\n$\\mathcal{V}_f\/\\mathcal{V}_f'\\cong\\mathbb{Z}^{k-1}\\oplus\\mathbb{Z}\/l\\mathbb{Z}$, where $l$ is the g.c.d.\\ of\n$l_1, l_2, \\ldots, l_k$.\n\nSuppose now that $\\deg f$ is odd. Then $\\mathcal{V}_f\/\\mathcal{V}_f'$ is isomorphic to\nthe quotient of $\\mathbb{Z}\/2\\mathbb{Z}\\oplus H_1(\\mathcal{M})$ by the relations\n$\\sigma_1(a)=a$, where $\\sigma_1(t, g)=(t+\\mathop{\\mathrm{sign}}(g),\n\\sigma(g))$, where $\\mathop{\\mathrm{sign}}(g)$ is the parity of the\nmonodromy action of $g$ for the covering map $f:\\mathcal{M}_1\\longrightarrow\\mathcal{M}$.\n\nIt follows that $\\sigma_1$ acts on the elements of the form $(0,\na_z)$, for $z\\in P_f$, by the rule\n\\[\\sigma_1(0, a_z)=\\left\\{\\begin{array}{ll} \\left(1, \\sum_{y\\in\n      f^{-1}(z)\\cap P_f}a_y\\right) & \\text{if $z$ is a critical value mod\n      2,}\\\\\n \\left(0, \\sum_{y\\in\n      f^{-1}(z)\\cap P_f}a_y\\right) &\n  \\text{otherwise.}\\end{array}\\right.\\]\n\nSuppose that $z\\in P_f$ is such that no point of\n$\\bigcup_{n\\ge 0}f^{-n}(z)$ is a critical value mod 2, and $z$ does not belong to a\ncycle. Then there exists $n$ such that $\\sigma_1^n(0, a_z)=0$, hence\nimage of $(0, a_z)$ in $\\mathcal{V}_f\/\\mathcal{V}_f'$ is zero.\n\nIf $z\\in P_f$ is a critical value mod 2, but no point\nof $\\bigcup_{n\\ge 1}f^{-n}(z)$ is a critical value mod 2, then\n$\\sigma_1(0, a_z)=(1, \\sigma(a_z))$, and hence the image of $(0, a_z)$ in $\\mathcal{V}_f\/\\mathcal{V}_f'$\nis equal to the image of $(1, 0)\\in\\mathbb{Z}\/2\\mathbb{Z}\\oplus H_1(\\mathcal{M})$.\n\nIt follows by induction that if $z\\in P_f$ does not belong to a cycle,\nthen the image of $(0, a_z)$ in $V_f\/\\mathcal{V}_f'$ is equal to the image of $(m,\n0)\\in\\mathbb{Z}\/2\\mathbb{Z}\\oplus H_1(\\mathcal{M})$, where $m$ is the parity of the number of\ncritical values mod 2 in the set $\\bigcup_{n\\ge 0}f^{-n}(z)$. In\nparticular, $\\mathcal{V}_f\/\\mathcal{V}_f'$ is a quotient of $\\mathbb{Z}\/2\\mathbb{Z}\\oplus H$, where\n$H\\le H_1(\\mathcal{M})$ is the subgroup generated by $a_z$ for $z$ belonging to cycles of\n$f:P_f\\longrightarrow P_f$.\n\nSuppose now that $C$ is a cycle of length $r$ of the map $f:P_f\\longrightarrow\nP_f$. For $z\\in C$, denote by $z'$ the unique element of $C$ such that\n$f(z')=z$, and by $t_z$ the parity of the number of\ncritical values mod 2 in the set\n$B_z=\\{z\\}\\cup\\bigcup_{y\\in f^{-1}(z)\\setminus z'}\\bigcup_{n\\ge 0}f^{-1}(y)$, see Figure~\\ref{fig:cycle}.\nThen $\\mathcal{V}_f\/\\mathcal{V}_f'$ is isomorphic to the quotient of $\\mathbb{Z}\/2\\mathbb{Z}\\oplus H$ by\nthe relations $(0, a_z)=(t_z, a_{z'})$.\n\n\\begin{figure}\n\\centering\n\\includegraphics{cycle.eps}\n\\caption{A post-critical cycle}\n\\label{fig:cycle}\n\\end{figure}\n\nNote that $t_C=\\sum_{z\\in C}t_z$ is the number of points $y$ that are\ncritical values mod 2 \nand $f^n(y)\\in C$ for all $n$ big enough. If $t_C$ is odd, then we\nhave a relation $(0, a_z)=(1, a_z)$, which implies that $(1, 0)$\nbelongs to the kernel of the epimorphism $\\mathbb{Z}\/2\\mathbb{Z}\\oplus\nH_1(\\mathcal{M})\\longrightarrow\\mathcal{V}_f\/\\mathcal{V}_f'$, and the arguments for even $\\deg f$ show that\n$\\mathcal{V}_f\/\\mathcal{V}_f'\\cong\\mathbb{Z}^{k-1}\\oplus\\mathbb{Z}\/l\\mathbb{Z}$. \n\nSuppose that $t_C$ is even for every cycle $C$ of $P_f$. Order\nelements of every cycle $C\\subset P_f$ into a sequence $z_0, z_1,\n\\ldots, z_{r-1}$ so that $f(z_i)=z_{i-1}$ for all $i=1, 2, \\ldots,\nr-1$, and $f(z_0)=z_{r-1}$. Denote then $b_{z_0}=(0, a_{z_0}),\nb_{z_1}=(t_{z_0}, a_{z_1}), b_{z_2}=(t_{z_0}+t_{z_1}, a_{z_2}),\n\\ldots, b_{z_{r-1}}=(t_{z_0}+t_{z_1}+\\cdots+t_{z_{r-2}},\na_{z_0})$. Then $\\mathcal{V}_f\/\\mathcal{V}_f'$ is the quotient of $\\mathbb{Z}\/2\\mathbb{Z}\\oplus H$ by\nthe relations $b_{z_i}=b_{z_{i+1}}$ and $b_{z_{r-1}}=b_{z_0}$. It\nfollows that $\\mathcal{V}_f\/\\mathcal{V}_f'$ is isomorphic to $\\mathbb{Z}\/2\\mathbb{Z}\\oplus\\mathbb{Z}^{k-1}\\oplus\\mathbb{Z}\/l\\mathbb{Z}$.\n\\end{proof}\n\n\\begin{example}\nIf $f(z)=z^2+c$ is a hyperbolic post-critically finite quadratic\npolynomial, then it has two\nattracting cycles: $\\{\\infty\\}$ and the orbit of the critical point\n0. It follows that $\\mathcal{V}_f\/\\mathcal{V}_f'$ is isomorphic to $\\mathbb{Z}$.\n\\end{example}\n\n\\begin{example}\nSuppose now that $f$ is a hyperbolic post-critically finite cubic\npolynomial. If it has only one finite critical point $c$, then\n$|f^{-1}(f(c))|=1$, hence there are no critical values mod 2. \n\nIf $f$ has two critical points $c_1, c_2$, then $f(c_1)$ and $f(c_2)$\nare critical values mod 2, and we have one of the following\npossibilities:\n\\begin{itemize}\n\\item[(a)] forward orbits of $c_1$ and $c_2$ are disjoint cycles;\n\\item[(b)] both points $c_1$ and $c_2$ belong to a common cycle;\n\\item[(c)] one of the critical points belongs to a cycle $C$, and the\nforward orbit of the other critical point eventually belongs to $C$.\n\\end{itemize}\n\nIt follows now from Proposition~\\ref{pr:hyperbolicrational}\nthat $\\mathcal{V}_f\/\\mathcal{V}_f'\\cong\\mathbb{Z}\/2\\mathbb{Z}\\oplus\\mathbb{Z}$ if the number of finite attracting\ncycles of $f$ is 1, and $\\mathcal{V}_f\/\\mathcal{V}_f'\\cong\\mathbb{Z}^2$ if it is 2.\n\\end{example}\n\n\\subsection{Finite presentation}\n\\begin{theorem}\n\\label{th:finitepresentation}\nIf $G$ is a contracting self-similar group, then the group $\\mathcal{V}_G$ is\nfinitely presented.\n\\end{theorem}\n\n\\begin{corollary}\nLet $f:\\mathcal{M}\\longrightarrow\\mathcal{M}$ be an expanding self-covering of a compact path\nconnected metric space. Then the group $\\mathcal{V}_f$ is finitely presented.\n\\end{corollary}\n\n\\begin{proof}\nLet $\\mathcal{N}$ be the nucleus of $G$. We may assume that $\\mathcal{N}$ is a\ngenerating set of $G$, since otherwise we can replace $G$ by\n$\\langle\\mathcal{N}\\rangle$ without changing $\\mathcal{V}_G$.\n\nFor $v\\in X^*$, and $g\\in\\mathcal{V}_G$,\ndenote by $L_v(g)$ the element of $\\mathcal{V}_G$ defined by the rule\n\\[L_v(g)(w)=\\left\\{\\begin{array}{cl} vg(u) & \\text{if $w=vu$ for some\n      $u\\in X^\\omega$}\\\\ w & \\text{if $w$ does not start with\n      $v$.}\\end{array}\\right.\\]\n\nThe following is straightforward.\n\n\\begin{proposition}\nFor every $v\\in X^*$ the map $L_v:\\mathcal{V}_G\\longrightarrow\\mathcal{V}_G$ is a group\nmonomorphism. If $v, u\\in X^*$ are not comparable, then the subgroups $L_v(\\mathcal{V}_G)$\nand $L_u(\\mathcal{V}_G)$ of $\\mathcal{V}_G$ commute. If $v, u\\in X^*$ are non-empty, and\n$h\\in\\mathcal{V}_G$ is such that $h(vw)=uw$ for all $w\\in X^\\omega$, then\n$h\\cdot L_v(g)\\cdot h^{-1}=L_u(g)$ for all $g\\in\\mathcal{V}_G$.\n\\end{proposition}\n\nFix a letter $x_1\\in X$. We will denote $L(g)=L_{x_1}(g)$.\nFor every pair $x, y\\in X$ choose elements $A_{x, y}$ and $B_x$ of $\\mathcal{V}_X$ such that\n\\[A_{x, y}(yw)=xyw,\\quad B_x(x_1w)=xw,\\]\nfor all $w\\in X^\\omega$. We assume that $B_{x_1}=1$.\n\nLet $\\langle S\\;|\\;R\\rangle$ be a finite presentation of the\nHigman-Thompson group $\\mathcal{V}_X$, see~\\cite{hgthomp}.\nLet $S_1$ be the set of elements of $\\mathcal{V}_G$ of the form $L(g)$ for\n$g\\in\\mathcal{N}$.\n\n\\begin{lemma}\nThe set $S\\cup S_1$ generates $\\mathcal{V}_G$.\n\\end{lemma}\n\n\\begin{proof}\nFor every non-empty $v\\in X^*$ we can find an element $h_v\\in\\mathcal{V}_X$ such that\n$h_v(vw)=x_1w$ for all $w\\in X^\\omega$, see\nLemma~\\ref{lem:incomplete}. Then $h_v^{-1}L(g)h_v=L_v(g)$\nfor all $g\\in\\mathcal{V}_G$. It follows that $L_v(g)\\in\\langle S\\cup\nS_1\\rangle$ for all $g\\in\\mathcal{N}$ and $v\\in X^*\\setminus\\{\\varnothing\\}$.\nEvery element $g\\in\\mathcal{V}_G$ can be\nrepresented by a table $\\left(\\begin{array}{cccc} v_1 & v_2 & \\ldots &\n    v_n\\\\ g_1 & g_2 & \\ldots & g_n\\\\ u_1 & u_2 & \\ldots &\n    u_n\\end{array}\\right)$, where $v_i, u_i$ are non-empty, and\n$g_i\\in\\mathcal{N}$. But then\n\\[g=\\left(\\begin{array}{cccc} v_1 & v_2 & \\ldots &\n    v_n\\\\ 1 & 1 & \\ldots & 1\\\\ u_1 & u_2 & \\ldots &\n    u_n\\end{array}\\right)L_{u_1}(g_1)L_{u_2}(g_2)\\cdots\n  L_{u_n}(g_n)\\in\\langle S\\cup S_1\\rangle.\\]\n\\end{proof}\n\nRepresent each $A_{x, y}$, $B_x$ as group words $\\overline A_{x, y}$,\n$\\overline B_x$ in $S$, and denote, for $y_1y_2\\ldots y_n\\in X^n$ and\n$y\\in X$,\n\\[\\overline A_{y_1y_2\\ldots y_n, y}=\n\\overline A_{y_1, y_2}\\cdots\\overline A_{y_{n-1}, y_n}\\overline A_{y_n, y}.\\]\nLet $A_{v, y}$ be the image of $\\overline A_{v, y}$ in $\\mathcal{V}_X$.\nThen $A_{v, y}$ satisfies\n\\[A_{v, y}(yw)=vyw\\]\nfor all $w\\in X^\\omega$.\n\nFor every word $v=y_1y_2\\ldots y_n$ of length at least 2 and every $g\\in\\mathcal{V}_G$ we have\n\\[L_v(g)= A_{y_1y_2\\ldots y_{n-1}, y_n}B_{y_n} L(g)B_{y_n}^{-1}\nA_{y_1y_2\\ldots y_{n-1}, y_n}^{-1}.\\]\nFor every $v=y_1y_2\\ldots y_n\\in X^*$ and $g\\in\\mathcal{N}$ denote by $\\overline L_v(g)$ the\nword\n\\[\\overline A_{y_1y_2\\ldots y_{n-1}, y_n}\\overline B_{y_n}\nL(g)\\overline B_{y_n}^{-1}\\overline A_{y_1y_2\\ldots y_{n-1}, y_n}^{-1}\\]\nin generators $S\\cup S_1$. Also denote by $\\overline L_x(g)$ the\nword $\\overline B_xL(g)\\overline B_x^{-1}$.\n\nDenote by $\\symm{d^n}$ the subgroup of $\\mathcal{V}_X$ consisting of all\nelements of the form $\\sum_{i=1}^{d^n}S_{u_i}S_{v_i}^*$,\nwhere $\\{v_1, v_2, \\ldots, v_{d^n}\\}=\\{u_1, u_2, \\ldots, u_{d^n}\\}=X^n$.\nIt is isomorphic to the symmetric group of degree $d^n$. Here $d=|X|$.\n\nFor every $x\\in X$ choose a finite generating set $W_x$ (as a\nset of group words in $S$) of the group $(\\mathcal{V}_X)_{(X^\\omega\\setminus\nxX^\\omega)}$ of elements of $\\mathcal{V}_X$ acting trivially on\n$xX^\\omega$. This group is isomorphic to the Higman-Thompson group\n$G_{d, d-1}$, hence is finitely generated (see~\\cite{hgthomp}).\n\nLet $R_1$ be the union of the following sets of relations.\n\\begin{itemize}\n\\item[(\\textbf{C})] \\textbf{Commutation.} Relations of the\n  form \\[[\\overline L_x(g_1), \\overline L_y(g_2)]=[\\overline L_{v_1}(g_1),\n \\overline L_{v_2}(g_2)]=[L(g), h]=1\\] for all $g, g_1, g_2\\in\\mathcal{N},\n x, y \\in X, v_1, v_2\\in X^2, h\\in W_{x_1}$, where $x\\ne y$ and\n $v_1\\ne v_2$.\n\n\\item[(\\textbf{N})] \\textbf{Nucleus.} Relations of the form \\[L(g_1)L(g_2)L(g_3)=1\\]\n  for all $x\\in X$ and $g_1, g_2, g_3\\in\\mathcal{N}$ such that $g_1g_2g_3=1$\n  in $G$.\n\n\\item[(\\textbf{S})] \\textbf{Splitting.} Relations of the form\n\\[L(g)=\\overline h\\cdot\\overline\n  L_{x_1y_1}(g|_{y_1})\\overline L_{x_1y_2}(g|_{y_2})\\cdots\\overline\n  L_{x_1y_d}(g|_{y_d}),\\] for all $g\\in\\mathcal{N}$, where $\\overline h$ is a word in the generators\n$S$ representing an element $h\\in\\symm{d^2}$ such that\n$L(g)=h L_{x_1y_1}(g|_{y_1})L_{x_1y_2}(g|_{y_2})\\cdots L_{x_1y_d}(g|_{y_d})$.\n\\end{itemize}\n\nLet us show that \nthe set $R\\cup R_1$ is a set of defining relations for the group $\\mathcal{V}_G$.\nDenote by $\\hat\\mathcal{V}_G$ the group defined by the presentation $\\langle\nS\\cup S_1\\;|\\;R\\cup R_1\\rangle$.\nAll relations $R\\cup R_1$ hold in $\\mathcal{V}_G$, hence $\\mathcal{V}_G$ is a quotient\nof $\\hat\\mathcal{V}_G$, and it is enough to show\nthat all relations in $\\mathcal{V}_G$ also hold in $\\hat\\mathcal{V}_G$.\n\nNote that since $R$ is a set of defining relations of $\\mathcal{V}_X$, a group\nword in $S$ is trivial in $\\hat\\mathcal{V}_G$ if\nand only if it is trivial in $\\mathcal{V}_X$. We will identify, therefore, the\nelements of the subgroup $\\langle S\\rangle\\le\\hat\\mathcal{V}_G$ with the corresponding elements of $\\mathcal{V}_X$.\n\n\\begin{lemma}\n\\label{lem:action}\nSuppose that $h\\in\\mathcal{V}_X$ and $u, v\\in X^*$ are such that $h(uw)=vw$ for\nall $w\\in X^\\omega$. Then $h\\overline L_u(g)h^{-1}=\\overline L_v(g)$\nholds in $\\hat\\mathcal{V}_G$.\n\\end{lemma}\n\n\\begin{proof}\nLet $u=a_1a_2\\ldots a_n$ and $v=b_1b_2\\ldots b_m$ for $a_i, b_i\\in\nX$. Then \\[\\overline L_u(g)=A_{a_1a_2\\ldots a_{n-1},\n  a_n}B_{a_n}L(g)B_{a_n}^{-1}A_{a_1a_2\\ldots a_{n-1}, a_n}^{-1}\\] and\n\\[\\overline L_v(g)=A_{b_1b_2\\ldots b_{m-1},\n  b_m}B_{b_m}L(g)B_{b_m}^{-1}A_{b_1b_2\\ldots b_{m-1}, b_m}^{-1},\\] by\ndefinition. We have dropped the lines above the letters $A$ and $B$,\nbecause the corresponding elements belong to $\\mathcal{V}_x$.\n\nWe have then\n\\begin{multline*}h\\overline L_u(g)h^{-1}=\nhA_{a_1a_2\\ldots a_{n-1},\n  a_n}B_{a_n}L(g)B_{a_n}^{-1}A_{a_1a_2\\ldots a_{n-1},\n  a_n}^{-1}h^{-1}=\\\\\nA_{b_1b_2\\ldots b_{m-1},  b_m}B_{b_m}\\cdot \nB_{b_m}^{-1}A_{b_1b_2\\ldots b_{m-1}, b_m}^{-1}hA_{a_1a_2\\ldots a_{n-1}, a_n}B_{a_n}\\cdot\\\\\nL(g)\\cdot\\\\\nB_{a_n}^{-1}A_{a_1a_2\\ldots a_{n-1}, a_n}^{-1}h^{-1}A_{b_1b_2\\ldots b_{m-1}, b_m}B_{b_m}\\cdot\nB_{b_m}^{-1}A_{b_1b_2\\ldots b_{m-1}, b_m}^{-1}.\n\\end{multline*}\n\nThe element \n\\[f=B_{b_m}^{-1}A_{b_1b_2\\ldots b_{m-1}, b_m}^{-1}hA_{a_1a_2\\ldots a_{n-1}, a_n}B_{a_n}\\]\nsatisfies\n\\[f(x_1w)=A_{v, y_m}^{-1}hA_{u, x_n}a_{y_m, x_n}(y_mw)=x_1w,\\] for all\n$w\\in X^\\omega$. Hence, by relations (\\textbf{C}), $L(g)$ commutes with $f$,\ni.e., $f\\cdot L(g)\\cdot f^{-1}=L(g)$ in\n$\\hat\\mathcal{V}_G$, which finishes the proof.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lem:Lcommutation}\nIf $v, u\\in X^*$ are incomparable, then $\\overline L_v(g_1)$ and\n$\\overline L_u(g_2)$ commute in $\\hat\\mathcal{V}_G$ for all $g_1, g_2\\in\\mathcal{N}$.\n\\end{lemma}\n\n\\begin{proof}\nLet $x, y\\in X$ be a pair of different letters. Then\n$[\\overline L_{xx}(g_1), \\overline L_{xy}(g_2)]=1$ in\n$\\hat\\mathcal{V}_G$, by (\\textbf{C}). Since $v, u$ are incomparable,\neither they both have length 1, or they form an incomplete\nantichain. In the first case commutation of $\\overline L_v(g_1)$ and\n$\\overline L_u(g_2)$ is a part of relations (\\textbf{C}). In the second case, there exists\n$a\\in\\mathcal{V}_X$ such that $a(uw)=xxw$ and $a(vw)=xyw$ for all\n$w$ (see Lemma~\\ref{lem:incomplete}). Then, by Lemma~\\ref{lem:action},\n\\[[\\overline L_v(g_1), \\overline L_u(g_2)]=a[\\overline\nL_{xx}(g_1), \\overline L_{xy}(g_2)]a^{-1}=1\\]\nin $\\hat\\mathcal{V}_G$.\n\\end{proof}\n\nLet us prove now that any group word in $S\\cup S_1$ that is trivial in\n$\\mathcal{V}_G$ is trivial in $\\hat\\mathcal{V}_G$. Note that relations (\\textbf{S})\nand Lemma~\\ref{lem:action} imply relations\n\\begin{itemize}\n\\item[(\\textbf{S'})] \\[\\overline L_v(g)=h\\cdot\\overline\n  L_{vy_1}(g|_{y_1})\\overline\n  L_{vy_2}(g|_{y_2})\\cdots\\overline L_{vy_d}(g|_{y_d})\\] for all\n  $g\\in\\mathcal{N}$ and non-empty $v\\in X^*$, where $h$ is an element of\n  $\\mathcal{V}_d$ such that $L_v(g)=hL_{vy_1}(g|_{y_1})L_{vy_2}(g|_{y_2})\\cdots\n  L_{vy_d}(g|_{y_d})$.\n\\end{itemize}\n\nEvery element of $\\hat\\mathcal{V}_G$ can be written in the form\n$hL(g_1)^{h_1}L(g_2)^{h_2}\\cdots L(g_n)^{h_n}$ for $h,\nh_i\\in\\mathcal{V}_X$, and $g_i\\in\\mathcal{N}$.\n\nLet $n_1$ be such that the element $h_n$ can be written as\n$\\sum_{i=1}^{d^{n_1}}S_{u_i}S_{v_i}^*$,\nwhere $\\{u_1, u_2, \\ldots, u_{d^{n_1}}\\}=X^{n_1}$. \nUsing relations\n(\\textbf{S'}) and Lemma~\\ref{lem:action}, we can rewrite the element\n$L(g_n)$ in the form $\\alpha\\prod_{v\\in X^{n_1-1}}\\overline\nL_{x_1v}(g_n|_v)$ for $\\alpha\\in\\symm{d^{n_1}}$. (Note that\nthe factors \n$\\overline L_{x_1v}(g_n|_v)$ commute with each other, by\nLemma~\\ref{lem:Lcommutation}.)\nThen for every $v\\in X^{n_1-1}$ there exists $i$ such that $x_1v=u_i$,\nand then by Lemma~\\ref{lem:action}, we have\n\\[{\\overline L_{x_1v}(g_n|_v)}^{h_n}=\\overline L_{v_i}(g_n|_v),\\]\nso that $L(g_n)^{h_n}$ can be rewritten as a product of\n$\\alpha^{h_n}$ followed by a product of elements\nof the form $\\overline L_v(g_{v, n})$ for some $v\\in X^{n_1}$ and\n$g|_{v, n}\\in\\mathcal{N}$.\n\nIt follows by induction that every element of $\\hat\\mathcal{V}_G$ can be\nwritten in the form\n\\begin{equation}\n\\label{eq:word}\ng=h\\overline L_{v_1}(g_1)\\overline L_{v_2}(g_2)\\cdots\\overline\nL_{v_m}(g_m)\n\\end{equation}\nfor some $v_i\\in X^*$, $g_i\\in\\mathcal{N}$, and $h\\in\\mathcal{V}_X$.\n\nSuppose that not all words $v_i$ are of the same length. Let $v_i$ be\nthe shortest, and let $k>|v_i|$ be the shortest length of words $v_j$\nstrictly longer than $v_i$. Using relations (\\textbf{S'}), we can\nrewrite $\\overline L_{v_i}(g_i)$ as $\\alpha_i\\prod_{u\\in\n  X^{k-|v_i|}}L_{v_iu}(g_i|_u)$, and then, using\nLemma~\\ref{lem:action}, move $\\alpha_i\\in\\symm{d^k}$ to the beginning of the\nproduct~\\eqref{eq:word}. This procedure will increase the length of\nthe shortest word $v_i$ in~\\eqref{eq:word} without changing the\nlength of the longest one. Repeating this procedure, we will\nchange~\\eqref{eq:word} to a product of the same form, but in which all\nwords $v_i$ are of the same length.\n\nTherefore, we may assume that in~\\eqref{eq:word} all words $v_i$ are\nof the same length $k$. Note that then $\\overline\nL_{v_1}(g_1)\\overline L_{v_2}(g_2)\\cdots\\overline L_{v_m}(g_m)$ does\nnot change the beginning of length $k$ in any word $w\\in\nX^\\omega$. Since $g$ is trivial in $\\mathcal{V}_G$, this implies that $h$ does\nnot change the beginning of length $k$ in any word. It follows that we\ncan write $h$ as a product $\\prod_{v\\in X^k}L_v(h_v)$ for some\n$h_v\\in\\mathcal{V}_X$. Using Lemmas~\\ref{lem:action} and~\\ref{lem:Lcommutation} we can\nnow rearrange the factors of~\\eqref{eq:word} in such a way that\n$g=\\prod_{v\\in X^k}f_v$, where $f_v=L_v(h_v)\\overline L_v(g_{v,\n  1})\\overline L_v(g_{v, 2})\\cdots\\overline L_v(g_{v, m_v})$ for\n$g_{v, i}\\in\\mathcal{N}$ and $h_v\\in\\mathcal{V}_X$. Note that $f_v$ are trivial in\n$\\mathcal{V}_G$. The latter implies that $h_v\\in\\symm{d^l}$ for some\n$l$, and that the action of $h_vg_{v, 1}g_{v, 2}\\cdots g_{v, m_v}$ on\n$X^l$ is trivial. Consequently, using relations (\\textbf{S'}), we can\nrewrite $f_v$ as a product of elements of the form $\\overline L_{vu}(g)$\nfor $u\\in X^l$. Therefore, we may assume that $h_v$ are trivial. Then\n$g_{v, 1}g_{v, 2}\\cdots g_{v, m_v}$ is trivial in $G$. Relations\n(\\textbf{N}), (\\textbf{S}), and\nPropositions~\\ref{pr:lengththree},~\\ref{pr:kernelcontracting} finish the proof.\n\\end{proof}\n\n\\section{Dynamical systems and groupoids}\n\nThis section is an overview of relations between expanding\ndynamical systems and self-similar groups, basic definitions of the\ntheory of \\'etale groupoids, and properties of hyperbolic groupoids. \nFor more details and proofs,\nsee~\\cite{nek:book,nek:filling,nek:models} \nand~\\cite{renault:groupoids,paterson:gr,haefl:foliations,nek:hyperbolic}.\n\n\\subsection{Limit dynamical system of a contracting group}\n\nLet $(G, X)$ be a contracting self-similar group. Denote by\n$X^{-\\omega}$ the space of all left-infinite sequences $\\ldots x_2x_1$\nof elements of $X$ with the direct product topology. \n\n\\begin{defi}\nSequences $\\ldots x_2x_1, \\ldots y_2y_1\\in X^{-\\omega}$ are\n\\emph{asymptotically equivalent} if there exists a finite set\n$N\\subset G$ and a sequence $g_n\\in N$\nsuch that \\[g_n(x_n\\ldots x_2x_1)=y_n\\ldots\ny_2y_1,\\] \nfor all $n\\in\\mathbb{N}$.\n\\end{defi}\n\nDenote by $\\mathcal{J}_G$ the quotient of the space $X^{-\\omega}$ by the\nasymptotic equivalence relation.\n\nThe asymptotic equivalence relation on $X^{-\\omega}$ is\ninvariant with respect to the shift $\\ldots x_2x_1\\mapsto\\ldots\nx_3x_2$. Therefore, the shift\ninduces a continuous map $f:\\mathcal{J}_G\\longrightarrow\\mathcal{J}_G$. The dynamical\nsystem $(f, \\mathcal{J}_G)$ is called the \\emph{limit dynamical system} of $G$.\n\nThe map $f:\\mathcal{J}_G\\longrightarrow\\mathcal{J}_G$ is expanding in the sense of\nDefinition~\\ref{def:expandingmap} (even though it is not a covering\nin general). Namely, we can represent $\\mathcal{J}_G$ as the boundary of a\nnaturally defined Gromov hyperbolic graph (see~\\cite{nek:hyplim}\nand~\\cite[Section~3.8]{nek:book}),\nand some iteration\nof $f$ will be locally uniformly expanding with respect to the visual\nmetric on the boundary.\n\n\\begin{defi}\n\\label{def:regular}\nA self-similar group $(G, X)$ is said to be \\emph{regular} if for every $g\\in G$ there\nexists a positive integer $n$ such that for every $v\\in X^n$ either\n$g(v)\\ne v$, or $g|_v=1$.\n\\end{defi}\n\nNote that it is enough to check the conditions of\nDefinition~\\ref{def:regular} for elements $g$ of the nucleus of $G$.\n\nThe following proposition is proved in~\\cite[Proposition~6.1]{nek:cpalg}.\n\n\\begin{proposition}\nThe shift map $f:\\mathcal{J}_G\\longrightarrow\\mathcal{J}_G$ is a covering if and only if $G$ is regular.\n\\end{proposition}\n\n\\begin{defi}\n\\label{def:selfreplicating}\nA self-similar action of $G$ on $X^*$ is said to be\n\\emph{self-replicating} (\\emph{recurrent} in~\\cite{nek:book})\nif the left action of $G$ on the associated\nbiset is transitive, i.e., if for every $x, y\\in X$ there exists $g\\in\nG$ such that $g\\cdot x=y\\cdot 1$.\n\nAn automorphism group $G$ of the rooted tree $X^*$ is said to be\n\\emph{level-transitive} if it is transitive on $X^n$ for every $n$.\n\\end{defi}\n\nNote that every self-replicating group is level transitive. \n\n\\begin{theorem}\nLet $G$ be a contracting group. The space $\\mathcal{J}_G$ is connected if and\nonly if $G$ is level-transitive. It is path connected if and only if\n$G$ is self-replicating. If $G$ is self-replicating, then $\\mathcal{J}_G$ is also\nlocally path connected.\n\\end{theorem}\n\n\\begin{proof}\nProof of connectedness, local connectedness, and path connectedness of\n$\\mathcal{J}_G$ under the appropriate conditions is given \nin~\\cite[Theorem~3.6.3.]{nek:book}. \n\nIf $\\ldots x_2x_1$ and $\\ldots y_2y_1$ are asymptotically equivalent\nelements of $X^{-\\omega}$, then for any $n$ the words $x_n\\ldots\nx_2x_1$ and $y_n\\ldots y_2y_1$ belong to the same $G$-orbit. If there\nexists $n$ such that the action of $G$ on $X^n$ is not transitive,\nthen partition of $X^n$ into $G$-orbits defines a partition of\n$X^{-\\omega}$ into clopen sets such that the asymptotic equivalence\nrelation identifies only points inside the sets of the partition. This\nimplies that $\\mathcal{J}_G$ is disconnected.\n\nThe same arguments shows that if $G$ is not self-replicating, then\n$\\mathcal{X}_G$ is disconnected. Moreover, if $v_1, v_2\\in X^n\\cdot G$ belong\nto different orbits of the left action, then for any $k, m\\ge 0$, and\nany $u_1, u_2\\in X^k\\cdot G$ and $w_1, w_2\\in X^m\\cdot G$ the elements\n$u_1\\otimes v_1\\otimes w_1$ and $u_2\\otimes v_2\\otimes w_2$ belong to\ndifferent orbits of the left action. It follows that the set of\nconnected components of $\\mathcal{X}_G$ is then uncountable. Consequently, the\nset of path connected components of $\\mathcal{X}_G$ is uncountable, and since\n$G$ is countable, the set of path connected components of $\\mathcal{J}_G=\\mathcal{X}_G\/G$ is\nalso uncountable.\n\\end{proof}\n\nThe following theorem is proved\nin~\\cite[Sections~5.3,~5.5]{nek:book} (in the context of length\nmetric spaces, but all the arguments remain to be valid in the general\ncase, if we use diameters of paths instead of their lengths, as in the\nproof of Proposition~\\ref{pr:expcontracting}.)\n\n\\begin{theorem}\nSuppose that $f:\\mathcal{J}\\longrightarrow\\mathcal{J}$ is an expanding self-covering of a path connected\nspace. Then $\\img{f}$ is contracting, regular, self-replicating,\nand the limit dynamical system of\n$\\img{f}$ is topologically conjugate to $(f, \\mathcal{J})$.\n\nLet $G$ be a contracting regular self-replicating group. Then it is\nequivalent, as a self-similar group (see\nDefinition~\\ref{def:ssequivalent}), to the iterated monodromy group\nof its limit dynamical system.\n\\end{theorem}\n\n\\begin{corollary}\nLet $f_i:\\mathcal{J}_i\\longrightarrow\\mathcal{J}_i$, for $i=1, 2$, be expanding self-coverings of\npath connected compact spaces. Then $(f_1, \\mathcal{J}_1)$ and $(f_2, \\mathcal{J}_2)$\nare topologically conjugate if and only if $\\img{f_1}$ and $\\img{f_2}$\nare equivalent as self-similar groups.\n\\end{corollary}\n\n\n\\subsection{Limit solenoid and the limit $G$-space}\n\\label{ss:solenoid}\n\nLet $X^{\\mathbb{Z}}$ be the space of all bi-infinite sequences $\\ldots\nx_{-2}x_{-1}.x_0x_1\\ldots$, where the dot denotes the place between the\ncoordinates number 0 and -1. \nSequences $(x_n)_{n\\in\\mathbb{Z}}, (y_n)_{n\\in\\mathbb{Z}}\\in X^\\mathbb{Z}$ are\n\\emph{asymptotically equivalent} if there exists a finite set\n$N\\subset G$ and a sequence $g_n\\in N$\nsuch that\n\\[g_n(x_nx_{n+1}\\ldots)=y_ny_{n+1}\\ldots\\] for all $n\\in \\mathbb{Z}$.\n\nThe quotient $\\mathcal{S}_G$ of $X^\\mathbb{Z}$ by the asymptotic equivalence\nrelation is called the \\emph{limit solenoid} of the group $G$. The\nshift $\\ldots x_{-2}x_{-1}.x_0x_1\\ldots\\mapsto\\ldots\nx_{-3}x_{-2}.x_{-1}x_0\\ldots$ induces a homeomorphism of\n$\\mathcal{S}_G$, which we will denote by $\\hat f$.\n\nIt is shown in~\\cite[Proposition~5.7.8.]{nek:book} that\nthe space $\\mathcal{S}_G$ is naturally homeomorphic to the inverse\nlimit of the backward iterations of the limit dynamical system $f:\\mathcal{J}_G\\longrightarrow\\mathcal{J}_G$:\n\\[\\mathcal{J}_G\\longleftarrow\\mathcal{J}_G\\longleftarrow\\mathcal{J}_G\\longleftarrow\\cdots,\\]\nand the map $\\hat f$ is conjugate to the map induced by $f$ on the inverse\nlimit. In other words, $(\\hat f, \\mathcal{S}_G)$ is the \\emph{natural\nextension} of the limit dynamical system $(f, \\mathcal{J}_G)$. The\npoint of $\\mathcal{S}_G$ represented by a sequence $\\ldots\nx_{-2}x_{-1}.x_0x_1\\ldots\\in X^{\\mathbb{Z}}$ corresponds to the point of the\ninverse limit represented by the sequence\n\\[\\cdots\\mapsto\\ldots x_{-2}x_{-1}x_0x_1x_2\\mapsto\n\\ldots x_{-2}x_{-1}x_0x_1\\mapsto\\ldots x_{-2}x_{-1}x_0\\mapsto\\ldots x_{-2}x_{-1}.\\]\n\n\nAnother natural dynamical system associated with a contracting group $G$\nis the limit $G$-space $\\mathcal{X}_G$. Consider the topological space $X^{-\\omega}\\times\nG$, where $G$ is discrete.\nTwo pairs $(\\ldots x_2x_1, g), (\\ldots y_2y_1, h)\\in X^{-\\omega}\\times\nG$ are \\emph{asymptotically equivalent} if there exists a sequence\n$g_n\\in G$ taking a finite set of values such that for all $n\\ge 1$\n\\[g_n\\cdot x_n\\ldots x_2x_1\\cdot g=y_n\\ldots y_2y_1\\cdot h\\]\nin the $n$th tensor power $\\Phi^{\\otimes n}$ of the associated\n$G$-biset, i.e., if\n\\[g_n(x_n\\ldots x_2x_1)=y_n\\ldots y_2y_1,\\qquad g_n|_{x_n\\ldots\n  x_2x_1}g=h.\\]\nThe quotient of $X^{-\\omega}\\times G$ by the asymptotic equivalence\nrelation is called the \\emph{limit $G$-space} $\\mathcal{X}_G$. We represent the points of the\nspace $\\mathcal{X}_G$ by the sequences $\\ldots x_2x_1\\cdot g$, where $x_i\\in X$\nand $g\\in G$.\n\nThe asymptotic equivalence relation on $X^{-\\omega}\\times G$ is\ninvariant with respect to the right action of $G$ on the second factor\nof the direct product. It follows that this action induces a right action\nof $G$ on $\\mathcal{X}_G$ by homeomorphisms. The action of $G$ on $\\mathcal{X}_G$ is\nproper, and the space of orbits $\\mathcal{X}_G\/G$ is naturally homeomorphic to\n$\\mathcal{J}_G$.\n\nThe spaces $\\mathcal{J}_G, \\mathcal{S}_G, \\mathcal{X}_G$ and the corresponding dynamical\nsystems depend only on the biset $\\Phi$ associated with the\nself-similar group. For example, $\\mathcal{X}_G$ can be constructed in the\nfollowing way. \n\nLet $\\Omega$ be the direct limit of the spaces $A^{-\\omega}$,\nwhere $A$ runs through all finite subsets of $\\Phi$. We write a\nsequence $(\\ldots, a_2, a_1)\\in A^{-\\omega}$ as $\\ldots\\otimes\na_2\\otimes a_1$. Two sequences $\\ldots \\otimes a_2\\otimes a_1,\n\\ldots\\otimes b_2\\otimes b_1\\in\\Omega$ are said to be equivalent if\nthere exist a sequence $g_n\\in G$ taking values in a finite set, such\nthat\n\\[g_n\\cdot a_n\\otimes \\cdots\\otimes a_2\\otimes a_1=b_n\\otimes\n\\cdots\\otimes b_2\\otimes b_1\\]\nin $\\Phi^{\\otimes n}$ for all $n$.\n\nThe quotient of $\\Omega$ by this equivalence relation is naturally\nhomeomorphic to $\\mathcal{X}_G$. Moreover, the homeomorphism conjugates the\nnatural action on $\\mathcal{X}_G$ with the action induced by the natural right\naction of $G$ on $\\Omega$.\n\nFor every $v\\cdot g\\in X^n\\cdot G=\\Phi^{\\otimes |v|}$ we have the map\n$w\\mapsto w\\otimes (v\\cdot g)$ on $\\Omega$, given in terms of\n$X^{-\\omega}\\times G$ by the rule\n\\[\\ldots x_2x_1\\cdot h\\mapsto\\ldots x_2x_1h(v)\\cdot h|_vg.\\]\nIt induces a continuous map $F_{v\\cdot g}:\\mathcal{X}_G\\longrightarrow\\mathcal{X}_G$. If $G$ is\nregular, then $F_{v\\cdot g}$ is a covering map.\n\nSince the limit dynamical system \n$f:\\mathcal{J}_G\\longrightarrow\\mathcal{J}_G$ is induced by the shift on $X^{-\\omega}$, the maps\n$F_{v\\cdot g}:\\mathcal{X}_G\\longrightarrow\\mathcal{X}_G$ are lifts of $f^{-|v|}$ by the quotient map\n$P:\\mathcal{X}_G\\longrightarrow\\mathcal{X}_G\/G=\\mathcal{J}_G$, i.e., we have equality $f^{|v|}\\circ P\\circ F_{v\\cdot g}=P$.\n\n\\subsection{Groupoids of germs}\n\\begin{defi}\nLet $\\mathcal{X}$ be a topological space. A \\emph{pseudogroup} acting on $\\mathcal{X}$\nis a set $\\widetilde\\mathfrak{G}$ of homeomorphisms between open subset of $\\mathcal{X}$ that is\nclosed under\n\\begin{enumerate}\n\\item \\emph{compositions}: if $F_1:U_1\\longrightarrow V_1$ and $F_2:U_2\\longrightarrow V_2$ are\n  elements of $\\widetilde\\mathfrak{G}$, then $F_1\\circ F_2:F_2^{-1}(V_2\\cap\n  U_1)\\longrightarrow F_1(V_2\\cap U_1)$ is an element of $\\widetilde\\mathfrak{G}$;\n\\item \\emph{taking inverse}: if $F:U\\longrightarrow V$ is an element of\n  $\\widetilde\\mathfrak{G}$, then $F^{-1}:V\\longrightarrow U$ is an element of $\\widetilde\\mathfrak{G}$;\n\\item \\emph{restriction} to an open subset: if $F:V\\longrightarrow U$ is an element of\n  $\\widetilde\\mathfrak{G}$ and $V'$ is an open subset of $V$, then $F|_{V'}\\in\\widetilde\\mathfrak{G}$;\n\\item \\emph{unions}: if for a homeomorphism $F:U\\longrightarrow V$ between open subsets\n  of $\\mathcal{X}$ there exists a covering $U_i$ of $U$ by open subsets, such\n  that $F|_{U_i}\\in\\widetilde\\mathfrak{G}$ for all $i$, then $F\\in\\widetilde\\mathfrak{G}$.\n\\end{enumerate}\nWe always assume that the identical homeomorphism $\\mathcal{X}\\longrightarrow\\mathcal{X}$ belongs\nto the pseudogroup.\n\\end{defi}\n\nLet $\\widetilde\\mathfrak{G}$ be a pseudogroup acting on $\\mathcal{X}$. A \\emph{germ}\nof $\\widetilde\\mathfrak{G}$ is equivalence class of \na pair $(F, x)$, where $F\\in\\widetilde\\mathfrak{G}$, and $x$ is a point of\nthe domain of $F$. Two pairs $(F_1, x_1)$ and $(F_2, x_2)$ represent\nthe same germ (are equivalent) if and only if $x_1=x_2$ and there\nexists a neighborhood $U$ of $x_1$ such that $F_1|_U=F_2|_U$.\n\nThe set of all germs of $\\widetilde\\mathfrak{G}$ has a natural topology whose\nbasis consists of sets of the form $\\{(F,\nx)\\;:\\;x\\in\\mathop{\\mathrm{dom}}(F)\\}$, where $F\\in\\widetilde\\mathfrak{G}$.\n\nIf $(F_1, x_1)$ and $(F_2, x_2)$ are such germs that $F_2(x_2)=x_1$,\nthen we can compose them:\n\\[(F_1, x_1)(F_2, x_2)=(F_1\\circ F_2, x_2).\\]\n\\emph{Inverse} of a germ $(F, x)$ is the germ $(F^{-1}, F(x))$. The\nset $\\mathfrak{G}$ of all germs of $\\widetilde\\mathfrak{G}$ is a groupoid with respect to these\noperations (i.e., a small category of isomorphisms).\n\nThe groupoid $\\mathfrak{G}$ is \\emph{topological}, i.e., the\noperations of composition and taking inverse\nare continuous.\n\n\\begin{example}\nIf $f:\\mathcal{X}\\longrightarrow\\mathcal{X}$ is a covering map, then restrictions of $f$ to open\nsubsets $U\\subset\\mathcal{X}$ such that $f:U\\longrightarrow f(U)$ is a homeomorphism\ngenerate a pseudogroup. Its groupoid of germs $\\mathfrak{F}$ will be called\n\\emph{groupoid of germs generated by $f$}. Every element of\n$\\mathfrak{F}$ can be represented as a product $(f^n, x)^{-1}(f^m, y)$\nfor some $x, y\\in\\mathcal{X}$ such that $f^m(y)=f^n(x)$.\n\\end{example}\n\n\\begin{example}\nIf $G$ is a group acting on a topological space $\\mathcal{X}$, then every germ\nof the pseudogroup generated by $G$ is a germ of an element of\n$G$. Therefore, the \\emph{groupoid of germs of $G$} is the set of\nequivalence classes of pairs $(g, x)\\in G\\times\\mathcal{X}$, where $(g_1, x_1)$\nand $(g_2, x_2)$ are equivalent if and only if $x_1=x_2$, and\n$g_1^{-1}g_2$ fixes all points of a neighborhood of $x_1$. This\ngroupoid is in general different from the \\emph{groupoid of the\naction}, which is equal as a set to $G\\times\\mathcal{X}$.\n\\end{example}\n\nIf $\\mathfrak{G}$ is a groupoid of germs of a pseudogroup acting on a space\n$\\mathcal{X}$, then we identify the germ $(1, x)$ of the identical\nhomeomorphism $1:\\mathcal{X}\\longrightarrow\\mathcal{X}$ with the point $x$ of $\\mathcal{X}$, and call\nelements of the form $(1, x)$ the \\emph{units} of the groupoid.\nWe will sometimes denote $\\mathcal{X}$ by $\\mathfrak{G}^{(0)}$, as the space of units of $\\mathfrak{G}$.\n\nFor $(F, x)\\in\\mathfrak{G}$, we denote by $\\mathsf{o}(F, x)=x$ and $\\mathsf{t}(F, x)=F(x)$ the\n\\emph{origin} and \\emph{target} of the germ. Two germs $g_1, g_2\\in\\mathfrak{G}$\nare \\emph{composable} (i.e., $g_1g_2$ is defined) if and only if\n$\\mathsf{t}(g_2)=\\mathsf{o}(g_1)$.\n\nWe say that points $x, y\\in\\mathfrak{G}^{(0)}$ \\emph{belong to one orbit} if\nthere exists $g\\in\\mathfrak{G}$ such that $x=\\mathsf{o}(g)$ and $y=\\mathsf{t}(g)$. This is an\nequivalence relation on $\\mathfrak{G}^{(0)}=\\mathcal{X}$, and this notion of orbits\ncoincides with the natural notion of orbits of pseudogroups.\nA set $A\\subset\\mathfrak{G}^{(0)}$ is a \\emph{$\\mathfrak{G}$-transversal} if it intersects\nevery $\\mathfrak{G}$-orbit.\n\nIf $A$ is a subset of $\\mathfrak{G}^{(0)}$, then \\emph{restriction} $\\mathfrak{G}|_A$ of $\\mathfrak{G}$ to\n$A$ is the groupoid of all elements $g\\in\\mathfrak{G}$ such that $\\mathsf{o}(g),\n\\mathsf{t}(g)\\in A$. The \\emph{isotropy} group $\\mathfrak{G}_x$, for $x\\in\\mathfrak{G}^{(0)}$,\nis the group of elements $g\\in\\mathfrak{G}$ such that $\\mathsf{o}(g)=\\mathsf{t}(g)=x$.\n\nNote that the pseudogroup $\\widetilde\\mathfrak{G}$ can be reconstructed from the\ngroupoid of its germs $\\mathfrak{G}$. Namely, a \\emph{bisection} is a\nsubset $F\\subset\\mathfrak{G}$ of the groupoid, such that\n$\\mathsf{o}:F\\longrightarrow\\mathsf{o}(F)$ and $\\mathsf{t}:F\\longrightarrow\\mathsf{t}(F)$ are homeomorphisms. Every open\nbisection $F$ defines a homeomorphism $\\mathsf{o}(F)\\longrightarrow\\mathsf{t}(F)$ by the rule\n$\\mathsf{o}(g)\\mapsto\\mathsf{t}(g)$ for $g\\in F$. The set $\\widetilde\\mathfrak{G}$ of all open\nbisections is a pseudogroup, and if $\\mathfrak{G}$ is the groupoid of germs of\na pseudogroup, then the pseudogroup of bisections coincides with $\\widetilde\\mathfrak{G}$. We say\nthat $\\widetilde\\mathfrak{G}$ is the \\emph{associated pseudogroup} of the groupoid.\n\n\\begin{defi}\n\\label{def:equivalentgroupoids}\nLet $\\mathfrak{G}_1, \\mathfrak{G}_2$ be groupoids of germs. We say that they are\n\\emph{equivalent} if there exists a groupoid $\\mathfrak{G}$ such that $\\mathfrak{G}^{(0)}$\nis the disjoint union $\\mathfrak{G}_1^{(0)}\\sqcup\\mathfrak{G}_2^{(0)}$, restrictions of\n$\\mathfrak{G}$ to $\\mathfrak{G}_i^{(0)}$ is equal to $\\mathfrak{G}_i$ for every $i=1, 2$, and the sets\n$\\mathfrak{G}_i^{(0)}$ are $\\mathfrak{G}$-transversals.\n\\end{defi}\n\nThe following procedure is a standard way of constructing a groupoid equivalent to a given\none. Namely, let\n$p:\\mathcal{Y}\\longrightarrow\\mathfrak{G}^{(0)}$ be a local homeomorphism, i.e., for every\n$y\\in\\mathcal{Y}$ there exists a neighborhood $U$ of\n$y$ such that $p:U\\longrightarrow p(U)$ is a homeomorphism. Suppose that\n$p(\\mathcal{Y})$ is a $\\mathfrak{G}$-transversal. Then \\emph{lift} of $\\mathfrak{G}$ by $p$ is\nthe groupoid of germs of the pseudogroup generated by all local\nhomeomorphisms of the form $p'\\circ F\\circ p:U\\longrightarrow W$, where \n\\begin{itemize}\n\\item $U$ is such that $p:U\\longrightarrow p(U)$ is a homeomorphism, \n\\item $p(U)$ is contained in the domain of $F$, \n\\item $W$ is such that $p:W\\longrightarrow F(p(U))$ is a homeomorphism, \n\\item $p'$ is the inverse of $p:W\\longrightarrow F(p(U))$.\n\\end{itemize}\nThen the map $p$ induces a morphism from the lift of $\\mathfrak{G}$ to $\\mathfrak{G}$,\nmapping the germ of $p'\\circ F\\circ p$ at $x$ to the germ of $F$ at $p(x)$.\n\n\n\\begin{example}\nConsider the \\emph{trivial groupoid} on a manifold $\\mathcal{M}$, i.e., the\ngroupoid consisting of units only. Let $\\pi:\\widetilde\\mathcal{M}\\longrightarrow\\mathcal{M}$ be the\nuniversal covering. Then lift of the trivial groupoid by $\\pi$ is the\ngroupoid of germs of the action of the fundamental group on\n$\\widetilde\\mathcal{M}$. (In this case it coincides with the groupoid of the action.)\n\\end{example}\n\n\\begin{defi}\nLet $\\mathfrak{G}$ be a groupoid of germs. It is said to be \\emph{proper} if the\nmap $(\\mathsf{o}, \\mathsf{t}):\\mathfrak{G}\\longrightarrow\\mathfrak{G}^{(0)}\\times\\mathfrak{G}^{(0)}$ is proper, i.e., if\npreimages of compact subsets of $\\mathfrak{G}^{(0)}\\times\\mathfrak{G}^{(0)}$ under this\nmap are compact.\n\\end{defi}\n\nThe groupoid $\\mathfrak{G}$ is proper if and only if for every compact subset\n$C$ of $\\mathfrak{G}^{(0)}$ the set of elements $g\\in\\mathfrak{G}$ such that $\\mathsf{o}(g),\n\\mathsf{t}(g)\\in C$ is compact.\n\nIf $\\mathfrak{G}$ is proper, then for every $x\\in\\mathfrak{G}^{(0)}$ the isotropy\ngroup $\\mathfrak{G}_x$ is finite.\n\nEvery groupoid equivalent to a proper groupoid is proper. If $\\mathfrak{G}$ is\nproper, then the space of orbits of $\\mathfrak{G}$ is Hausdorff.\n\nLet $\\mathfrak{G}$ be a groupoid of germs. Its \\emph{topological full group}\n$[[\\mathfrak{G}]]$ is the set of all bisections $F$ such that\n$\\mathsf{o}(F)=\\mathsf{t}(F)=\\mathfrak{G}^{(0)}$, i.e., the set \nof homeomorphisms $F:\\mathfrak{G}^{(0)}\\longrightarrow\\mathfrak{G}^{(0)}$ such that all germs\nof $F$ belong to $\\mathfrak{G}$. See~\\cite{gior:full}, where the notion of a topological\nfull group (for a groupoid of germs generated by one homeomorphism)\nwas introduced.\n\n\\begin{example}\nLet $f:\\mathcal{M}_1\\longrightarrow\\mathcal{M}$ be a partial self-covering. Then $\\mathcal{V}_f$ is the full\ntopological group of the groupoid of germs of the local homeomorphisms\n$S_\\gamma$ of the boundary of the tree $T_t$ for $t\\in\\mathcal{M}$.\n\\end{example}\n\n\\begin{example}\nLet $G$ be a self-similar group acting on $X^\\omega$. Let $\\mathfrak{G}$ be the\ngroupoid of germs of the pseudogroup generated by the action of $G$\nand the germs of the homeomorphisms $S_x(x_1x_2\\ldots)=xx_1x_2\\ldots$\nfor $x\\in X$. It is easy to see that the topological full group of\n$\\mathfrak{G}$ is the group $\\mathcal{V}_G$.\n\\end{example}\n\n\\subsection{Hyperbolic groupoids}\n\nHere we present a very short overview of the basic definitions and\nresults of the paper~\\cite{nek:hyperbolic}.\n\nLet $\\mathfrak{G}$ be a groupoid of germs. A \\emph{compact generating pair} of $\\mathfrak{G}$ is a\npair $(S, \\mathcal{X}_1)$, where $S\\subset\\mathfrak{G}$ and $\\mathcal{X}_1\\subset\\mathfrak{G}^{(0)}$ are\ncompact, $\\mathcal{X}_1$ contains an open $\\mathfrak{G}$-transversal,\n$S\\subset\\mathfrak{G}|_{\\mathcal{X}_1}$, and for every\n$g\\in\\mathfrak{G}|_{\\mathcal{X}_1}$ there exists $n$ such that $(S\\cup S^{-1})^n$ is a\nneighborhood of $g$ in $\\mathfrak{G}|_{\\mathcal{X}_1}$.\n\nA groupoid is \\emph{compactly generated} if it has a compact\ngenerating pair. See a variant of this definition\nin~\\cite{haefliger:compactgen}. A groupoid equivalent to a compactly generated\ngroupoid is also compactly generated. \n\nLet $(S, \\mathcal{X}_1)$ be a compact generating pair of $\\mathfrak{G}$. Let\n$x\\in\\mathcal{X}_1$. Then the \\emph{Cayley graph} $\\Gamma(x, S)$ is the\noriented graph with the set of vertices \\[\\{g\\in\\mathfrak{G}\\;:\\;\\mathsf{o}(g)=x,\n\\mathsf{t}(g)\\in\\mathcal{X}_1\\},\\] in which there is an arrow from $g_1$ to $g_2$ if\nand only if $g_2g_1^{-1}\\in S$.\n\nA vertex path $v_1, v_2, \\ldots$ \nin a graph $\\Gamma$ (i.e., a sequences of vertices such that $v_i$ is\nadjacent to $v_{i+1}$) is said to be a $C$-quasi-geodesic (where $C>1$ is\na constant) if $|v_i-v_j|\\ge C^{-1}|i-j|+C$ for all $i, j$.\n\n\\begin{defi}\n\\label{def:hyperbolic}\nA Hausdorff groupoid of germs $\\mathfrak{G}$ is \\emph{hyperbolic} if there is\na compact generating pair $(S, \\mathcal{X}_1)$ of $\\mathfrak{G}$, a metric $|x-y|$\ndefined on a neighborhood of $\\mathcal{X}_1$, and constants $L, C>1, \\Delta>0$\nsuch that\n\\begin{enumerate}\n\\item Every element $g\\in S$ is a germ of a homeomorphism $F\\in\\widetilde\\mathfrak{G}$ such\n  that $|F(x)-F(y)|\\le L^{-1}|x-y|$ for all $x, y\\in\\mathop{\\mathrm{Dom}} F$.\n\\item For every $x\\in\\mathcal{X}_1$ the Cayley graph $\\Gamma(x, S)$ is Gromov\n  $\\Delta$-hyperbolic.\n\\item For every $x\\in\\mathcal{X}_1$ there exists a point $\\omega_x$ of the\n  boundary of $\\Gamma(x, S)$ such that every oriented path in the\n  Cayley graph $\\Gamma(x, S^{-1})$ is a $C$-quasi-geodesic converging\n  to $\\omega_x$.\n\\item $\\mathsf{o}(S)=\\mathsf{t}(S)=\\mathcal{X}_1$.\n\\item All elements of the pseudogroup $\\widetilde\\mathfrak{G}$.\n\\end{enumerate}\n\\end{defi}\n\n\\begin{example}\nLet $f:\\mathcal{J}\\longrightarrow\\mathcal{J}$ be an expanding self-covering of a compact metric space. Then the groupoid of\ngerms generated by $f$ is hyperbolic. The corresponding generating\npair is $(S, \\mathcal{J})$, where $S$ is the set of germs of $f^{-1}$. The\ncorresponding Cayley graphs $\\Gamma(x, S)$ are trees. The special\npoint $\\omega_x$ of the boundary is the limit of the forward germs\n$(f^n, x)$ for $n\\to+\\infty$. See more in~\\cite[Section~5.2]{nek:hyperbolic}\n\\end{example}\n\nLet $\\mathfrak{G}$ be a hyperbolic groupoid. For every $x\\in\\mathfrak{G}^{(0)}$ there\nexists a generating pair $(S, \\mathcal{X}_1)$ satisfying the conditions of\nDefinition~\\ref{def:hyperbolic}\nand such that $x\\in\\mathcal{X}_1$. Denote by $\\partial\\mathfrak{G}_x$ the boundary of the\nCayley graph $\\Gamma(x, S)$ minus the point $\\omega_x$. The space\n$\\partial\\mathfrak{G}_x$ does not depend on the generating pair.\n\nLet $(S, \\mathcal{X}_1)$ be a generating pair satisfying the conditions of\nDefinition~\\ref{def:hyperbolic}. Find a finite set of contractions\n$\\mathcal{F}\\subset\\tilde\\mathfrak{G}$ such that $S\\subset\\bigcup_{F\\in\\mathcal{F}} F$, i.e.,\nevery element $s\\in S$ is a germ of a contraction $F\\in\\mathcal{F}$. Every point\n$\\xi\\in\\partial\\mathfrak{G}_x$ can be represented as the limit of a sequence\nvertices of the Cayley graph of\nthe form $g, s_1g, s_2s_1g, \\ldots, s_n\\cdots s_2s_1g, \\ldots$, where\n$g\\in\\mathfrak{G}$, and $s_i\\in S$.\nThere exists $\\epsilon>0$ (not depending on $\\xi$) and a sequence\n$F_i\\in\\mathcal{F}$ such that $s_i$ is equal to a germ $(F_i, x_i)$, and the\n$\\epsilon$-neighborhood of $x_i=\\mathsf{o}(s_i)$ belongs to the domain of\n$F_i$. Then there exists $\\delta$ (also depending only on $S$ and\n$\\mathcal{F}$) such that the $\\delta$-neighborhood of $\\mathsf{t}(g)$ belongs to\nthe domain of $F_n\\circ\\cdots\\circ F_2\\circ F_1$ for all\n$n$.\n\nLet $F\\in\\widetilde\\mathfrak{G}$ be such that $g\\in F$ and $\\mathsf{t}(F)$ belongs to the\n$\\delta$-neighborhood of $\\mathsf{t}(g)$. Then $\\mathsf{o}(F_n\\circ\\cdots\\circ\nF_2\\circ F_1\\circ F)=\\mathsf{o}(F)$ for every $n$.\n\nIt is shown in~\\cite{nek:hyperbolic} that there exists a topology on\nthe disjoint union $\\partial\\mathfrak{G}$ of the spaces $\\partial\\mathfrak{G}_x$,\n$x\\in\\mathfrak{G}^{(0)}$, which agrees with the topology on its subsets\n$\\partial\\mathfrak{G}_x$ and such that the map\n\\begin{equation}\n\\label{eq:locprod}\n(y, \\xi)\\mapsto \\lim_{n\\to\\infty}(F_n\\circ\nF_{n-1}\\circ\\cdots\\circ F_1\\circ F, y)\\in\\partial\\mathfrak{G}_y\n\\end{equation}\nis a well defined homeomorphism (if $U=\\mathsf{o}(F)$ is small enough)\nfrom the direct product of $U=\\mathsf{o}(F)$\nwith a subset of $\\partial\\mathfrak{G}_x$ to a subset of $\\partial\\mathfrak{G}$, see\nFigure~\\ref{fig:tube}.\n\n\\begin{figure}\n\\centering\n\\includegraphics{tube.eps}\n\\caption{Composition of contractions}\n\\label{fig:tube}\n\\end{figure}\n\nMoreover, these homeomorphisms agree with a natural \\emph{local product\nstructure} of $\\partial\\mathfrak{G}$. Namely, a basis of the topology on\n$\\partial\\mathfrak{G}$ consists of \\emph{rectangles}, i.e., sets with a\ndecomposition into a direct product of topological spaces, such that\nthe decompositions agree where they overlap, and locally coincide with\nthe maps given by~\\eqref{eq:locprod}.\n\nThe groupoid $\\mathfrak{G}$ acts on the space $\\partial\\mathfrak{G}$ from the\nright. Namely, every $g\\in\\mathfrak{G}$ defines a natural homeomorphism\n$\\partial\\mathfrak{G}_{\\mathsf{t}(g)}\\longrightarrow\\partial\\mathfrak{G}_{\\mathsf{o}(g)}$ mapping the limit of a\nsequence $g_n\\in\\Gamma(\\mathsf{t}(g), S)$ to the limit of the sequence\n$g_ng\\in\\Gamma(\\mathsf{o}(g), S)$. This action is an action of the topological\ngroupoid $\\mathfrak{G}$ on the topological space $\\partial\\mathfrak{G}$ over the projection map\n$P:\\partial\\mathfrak{G}\\longrightarrow\\mathfrak{G}^{(0)}$ mapping all points of $\\partial\\mathfrak{G}_x$ to\n$x$.\n\nThe action of $\\mathfrak{G}$ on $\\partial\\mathfrak{G}$ (i.e., the associated action of\n$\\tilde\\mathfrak{G}$ on $\\partial\\mathfrak{G}$ by local homeomorphisms) \npreserves with the local product structure of $\\partial\\mathfrak{G}$. Naturally\ndefined projection of the action of $\\mathfrak{G}$ onto the first coordinate of\nthe local product decomposition is equivalent to $\\mathfrak{G}$, while the projection\nonto the second coordinate is the \\emph{dual groupoid} of $\\mathfrak{G}$.\n\nLet us give an equivalent, and maybe more intuitive, definition of the dual groupoid.\n\n\\begin{defi}\nLet $\\overline{\\Gamma(x, S)}$ and $\\overline{\\Gamma(y, S)}$ be the\nCayley graphs of $\\mathfrak{G}$ with adjoined boundaries $\\partial\\mathfrak{G}_x$ and\n$\\partial\\mathfrak{G}_y$. A homeomorphism $F:U\\longrightarrow V$ between open neighborhoods\n$U\\subset\\overline{\\Gamma(x, S)}$ and $V\\subset\\overline{\\Gamma(y,\n  S)}$ of points of $\\partial\\mathfrak{G}_x$ and $\\partial\\mathfrak{G}_y$ is an\n\\emph{asymptotic morphism} if for every sequence of pairwise different\nedges $(g_1, h_1), (g_2, h_2), \\ldots$ in $U$ the distance between\n$g_ih_i^{-1}$ and $F(g_i)F(h_i)^{-1}$ goes to zero.\n\\end{defi}\n\nNote that $g_ih_i^{-1}$ and $F(g_i)F(h_i)^{-1}$ belong to a compact\nsubset of $\\mathfrak{G}$, hence the notion of convergence of their distance to\nzero does not depend on the choice of a metric on $\\mathfrak{G}$.\n\n\\begin{defi}\nThe groupoid $\\mathfrak{d}\\mathfrak{G}$ of germs of restrictions of the asymptotic morphisms \nto the spaces $\\partial\\mathfrak{G}_x$, $x\\in\\mathfrak{G}^{(0)}$, is\nthe \\emph{dual groupoid} of $\\mathfrak{G}$.\n\\end{defi}\n\nThe space of units of $\\mathfrak{d}\\mathfrak{G}$ is the topologically disjoint union of\nthe spaces $\\partial\\mathfrak{G}_x$. (In particular, it is not separable.)\nIf $\\mathfrak{G}$ is minimal (i.e., if all orbits are dense),\nthen $\\partial\\mathfrak{G}_x$ is an open transversal of the\ndual groupoid for any $x\\in\\mathfrak{G}^{(0)}$,\nhence the dual groupoid can be defined as the groupoid\nof germs at $\\partial\\mathfrak{G}_x$ of the asymptotic morphisms.\nWe will denote it $\\mathfrak{d}\\mathfrak{G}_x$.\n\nWe will denote by $\\mathfrak{G}^\\top$ any groupoid equivalent to $\\mathfrak{d}\\mathfrak{G}$.\nThe following theorem is proved in~\\cite{nek:hyperbolic}.\n\n\\begin{theorem}\nLet $\\mathfrak{G}$ be a minimal Hausdorff hyperbolic groupoid. Then the dual\ngroupoid $\\mathfrak{G}^\\top$ is minimal, Hausdorff, and hyperbolic, and\n$(\\mathfrak{G}^\\top)^\\top$ is equivalent to $\\mathfrak{G}$.\n\\end{theorem}\n\n\\subsection{Groupoid of germs generated by an expanding self-covering}\n\nLet $f:\\mathcal{J}\\longrightarrow\\mathcal{J}$ be an expanding self-covering of a path connected\ncompact metric space. Denote by $\\mathfrak{F}$ the groupoid of germs\ngenerated by $f$. Every element of $\\mathfrak{F}$ can be written as a\nproduct $(f^n, x)^{-1}(f^m, y)$, for $n, m\\in\\mathbb{N}$, and $x, y\\in\\mathcal{J}$ such\nthat $f^n(x)=f^m(y)$.\n\nA \\emph{natural extension} of $f$ is the inverse limit $\\hat\\mathcal{J}$ of the\nmaps $f$ together with the homeomorphism $\\hat f$ of $\\hat\\mathcal{J}$ induced\nby $f$, see~\\ref{ss:solenoid}. Let $P_S:\\hat\\mathcal{J}\\longrightarrow\\mathcal{J}$ be the natural projection.\n\nFor every point $x\\in\\mathcal{J}$ there exists a neighborhood $U$ that is\nevenly covered by each map $f^n:\\mathcal{J}\\longrightarrow\\mathcal{J}$, since $f$ is expanding. It\nfollows that the set $P^{-1}(U)$ is naturally decomposed into the\ndirect product of $U$ with the boundary $\\partial T_x$ of the tree of\npreimages of any point $x\\in U$. \n\nThe groupoid $\\mathfrak{F}$ is hyperbolic, and we can\nconsider the space $\\partial\\mathfrak{F}$ together with the projection\n$P:\\partial\\mathfrak{F}\\longrightarrow\\mathfrak{F}^{(0)}=\\mathcal{J}$.\nLet us use the generating set $S$ of\n$\\mathfrak{F}$ equal to the set of germs of the inverse map\n$f^{-1}$. Then the Cayley graphs $\\Gamma(x, S)$ are regular trees such\nthat every vertex has one incoming and $d=\\deg f$ outgoing arrows. The\nfiber $\\partial\\mathfrak{F}_x$ is equal to the boundary of this tree\nminus the limit of the path $(f^n, x)$, $n\\ge 0$. In other words, it\nis the natural inductive limit of the boundaries of the preimage\ntrees $T_{f^n(x)}$ for $n\\ge 0$.\n\nLet $\\nu:\\mathfrak{F}\\longrightarrow\\mathbb{Z}$ be the homomorphism (\\emph{cocycle})\ngiven by the rule $\\nu(f, x)=-1$, so that $\\nu((f^n, x)^{-1}(f^m,\ny))=n-m$. See Figure~\\ref{fig:cayleye}, where the Cayley graph $\\Gamma(x, S)$\ntogether with the levels of the cocycle $\\nu$ are shown.\n\n\\begin{figure}\n\\centering\n\\includegraphics{cayleye.eps}\n\\caption{Cayley graph of $\\mathfrak{F}$}\n\\label{fig:cayleye}\n\\end{figure}\n\nEvery point of $\\partial\\mathfrak{F}_x$ can be uniquely represented as the\nlimit of a sequence $s_n\\cdots s_2s_1\\cdot g$, where $s_i\\in S$, and\n$g\\in\\mathfrak{F}$ is such that $\\mathsf{o}(g)=x$ and $\\nu(g)=0$. Note that\nthe set of limits of the sequences $s_n\\cdots s_2s_1\\cdot g$ for a\nfixed $g$ and all possible choices of $s_i\\in S$ is\nnaturally identified with the fiber $P_S^{-1}(\\mathsf{t}(g))$ of the solenoid\n$\\hat\\mathcal{J}$ (i.e., with $\\partial T_{\\mathsf{t}(g)}$). It\nfollows then directly from the definitions that the space\n$\\partial\\mathfrak{F}$ is homeomorphic to the subset \n$\\{(\\zeta, g)\\;:\\;P_S(\\zeta)=\\mathsf{t}(g)\\}$\nof the direct product $\\hat\\mathcal{J}\\times\\mathfrak{F}_0$, \nwhere $\\mathfrak{F}_0$ is the subgroupoid\n$\\nu^{-1}(0)\\subset\\mathfrak{F}$. \nThe action of\n$\\mathfrak{F}$ on $\\partial\\mathfrak{F}$ is given in these terms by the rules\n\\[(\\zeta, g)\\cdot h=\\left\\{\\begin{array}{ll}({\\hat f}^n(\\zeta),\n    f^n\\circ gh), & \\text{if $\\nu(h)=n>0$,}\\\\\n({\\hat f}^{-n}(\\zeta), s_n\\cdots s_2s_1gh), & \\text{if $\\nu(h)=-n<0$,}\\\\\n(\\zeta, gh), &\\text{if $\\nu(h)=0$,}\n\\end{array}\\right.\\]\nwhere $s_i\\in S$ are such that $\\zeta=\\lim_{m\\to\\infty}s_m\\cdots\ns_2s_1$.\n\nWe get hence the following description of the natural extension $\\hat\nf:\\hat\\mathcal{J}\\longrightarrow\\hat\\mathcal{J}$ in terms of the $\\mathfrak{F}$-space\n$\\partial\\mathfrak{F}$.\n\n\\begin{proposition}\n\\label{pr:solenoidflow}\nThe quotient of the space $\\partial\\mathfrak{F}$ by the action of\n$\\mathfrak{F}_0=\\nu^{-1}(0)$ is homeomorphic to $\\hat\\mathcal{J}$. If\n$F\\in\\tilde{\\mathfrak{F}}$ is such that $\\nu(F)=\\{n\\}$, then the\ngerms of the map induced by $F$ on the quotient space\n$\\hat\\mathcal{J}=\\partial\\mathfrak{F}\/\\mathfrak{F}_0$ are germs of the map\n${\\hat f}^{-n}$.\n\\end{proposition}\n\nWe say that points $\\xi, \\zeta\\in\\hat\\mathcal{J}$ are\n\\emph{unstably equivalent} if distance between\n$\\hat f^{-n}(\\xi)$ and $\\hat f^{-n}(\\zeta)$ goes to zero as $n\\to\n+\\infty$. They are said to be \\emph{stably equivalent} if distance between\n$\\hat f^n(\\xi)$ and $\\hat f^n(\\zeta)$ goes to zero as $n\\to +\\infty$.\n\nA \\emph{leaf} of $\\hat\\mathcal{J}$ is its path connected component. (Recall\nthat we assume that $\\mathcal{J}$ is path connected.) Every leaf \nis an equivalence class of the unstable\nequivalence relation on $\\hat\\mathcal{J}$. \n\nEach leaf is dense in $\\hat\\mathcal{J}$, and it is more natural to consider it\nwith the \\emph{inductive limit topology}.\nNamely, a subset $U$ of a leaf $\\mathcal{L}$ is open if and only if\nits intersection with every compact subset $C\\subset\\mathcal{L}$ is\nopen in $C$. Note that for every compact set $C\\subset\\mathcal{L}$ the\nmap $P:C\\longrightarrow\\mathcal{J}$ is finite-to-one.\n\nRestriction of the map\n$P:\\hat\\mathcal{J}\\longrightarrow\\mathcal{J}$ to any leaf $\\mathcal{L}$ of $\\hat\\mathcal{J}$ is a covering\nmap. Let $\\mathfrak{F}_{\\mathcal{L}}$ be the lift of the groupoid\n$\\mathfrak{F}$ to the leaf $\\mathcal{L}$ by this covering. Then the\ngroupoid $\\mathfrak{F}_{\\mathcal{L}}$ is equivalent to\n$\\mathfrak{F}$. The cocycle $\\nu$ lifts to the cocycle $\\nu\\circ P$ on\n$\\mathfrak{F}_{\\mathcal{L}}$, which we will denote by\n$\\nu_{\\mathcal{L}}$ or just $\\nu$.\n\nIt follows then from the definition of $\\partial\\mathfrak{G}$ for a hyperbolic\ngroupoid $\\mathfrak{G}$, that \nthe space $\\partial\\mathfrak{F}_{\\mathcal{L}}$ is the fiber product of\nthe maps $P_S:\\mathcal{L}\\longrightarrow\\mathcal{J}$ and $P:\\partial\\mathfrak{F}\\longrightarrow\\mathcal{J}$,\ni.e., the subset $\\{(x, y)\\;:\\;P_S(x)=P(y)\\}$ of\n$\\mathcal{L}\\times\\partial\\mathfrak{F}$. We get the following\ncorollary of Proposition~\\ref{pr:solenoidflow}.\n\n\\begin{corollary}\n\\label{cor:solenoidflow}\nLet $\\mathcal{L}$ be a leaf of $\\hat\\mathcal{J}$, and let\n$\\mathfrak{F}_{\\mathcal{L}}$ be the lift of $\\mathfrak{F}$ by the\ncovering $P_S:\\mathcal{L}\\longrightarrow\\mathcal{J}$.\nThen the quotient of the space $\\partial\\mathfrak{F}_{\\mathcal{L}}$ by the action of\n$\\mathfrak{F}_{\\mathcal{L}, 0}=\\nu_{\\mathcal{L}}^{-1}(0)$ is\nhomeomorphic to $\\hat\\mathcal{J}$. If $F\\in\\tilde{\\mathfrak{F}_{\\mathcal{L}}}$\nis such that $\\nu_{\\mathcal{L}}(F)=\\{n\\}$, then the\ngerms of the map induced by $F$ on the quotient space\n$\\hat\\mathcal{J}=\\partial\\mathfrak{F}_{\\mathcal{L}}\/\\mathfrak{F}_{\\mathcal{L},\n0}$ are germs of the map ${\\hat f}^{-n}$.\n\\end{corollary}\n\nLet $G$ be a contracting regular self-replicating group. Let $\\mathfrak{G}$ be\nthe groupoid of\ngerms of the action of the group $\\mathcal{V}_G$ on $X^\\omega$. It is generated\nby the groupoid of germs of $G$ and the germs of the maps\n$S_x:x_1x_2\\ldots\\mapsto xx_1x_2\\ldots$. It is shown\nin~\\cite[Subsection~5.3.]{nek:hyperbolic} that $\\mathfrak{G}$\nis hyperbolic, and its dual is the groupoid generated by the limit\ndynamical system $f:\\mathcal{J}_G\\longrightarrow\\mathcal{J}_G$.\n\nMore explicitly, let $w\\in X^\\omega$. Then the boundary\n$\\partial\\mathfrak{G}_w$ is the leaf of the limit solenoid $\\mathcal{S}_G$\nconsisting of points representable by sequences $\\ldots\nx_{-2}x_{-1}\\cdot x_0x_1\\ldots$, where $x_0x_1\\ldots$ belongs to the\n$G$-orbit of $w$. The groupoid $\\mathfrak{d}\\mathfrak{G}_w$ is equal to the\nlift of the groupoid generated by the limit dynamical system\n$f:\\mathcal{J}_G\\longrightarrow\\mathcal{J}_G$ to the leaf $\\partial\\mathfrak{G}_w$ (by the covering induced by the \nprojection $\\mathcal{S}_G\\longrightarrow\\mathcal{J}_G$ of the natural extension onto $\\mathcal{J}_G$).\n\n\n\\section{Reconstruction of the dynamical system from $\\mathcal{V}_f$}\n\nThe main result of this section is the following classification of the\ngroups $\\mathcal{V}_f$.\n\n\\begin{theorem}\n\\label{th:classification}\nLet $f_i:\\mathcal{J}_i\\longrightarrow\\mathcal{J}_i$, for $i=1, 2$, be expanding self-coverings of\npath connected compact metric spaces. Then $\\mathcal{V}_{f_1}$ and $\\mathcal{V}_{f_2}$\nare isomorphic as abstract groups if and only if the dynamical systems\n$(f_1, \\mathcal{J}_1)$ and $(f_2, \\mathcal{J}_2)$ are topologically conjugate.\n\\end{theorem}\n\n\\begin{example}\nOne can show that the limit dynamical systems of two groups\n$\\mathfrak{K}_{v_i}$, $i=1, 2$, are topologically conjugate if and\nonly if either $v_1=v_2$, or $v_1$ can be obtained from $v_2$ by\nreplacing each 0 by 1 and each 1 by 0. Namely, the sequence of letters\nof $v_i$ can be interpreted as a \\emph{kneading sequence} of the\ndynamical system, which in turn can be defined in purely topological\nterms. This gives a complete classification of the groups\n$\\mathcal{V}_{\\mathfrak{K}_v}$ up to isomorphism.\n\\end{example}\n\n\\subsection{M.~Rubin's theorem}\n\nRecall that if $G$ is a group acting on a topological space $\\mathcal{X}$, and\n$U\\subseteq\\mathcal{X}$ is an open subset, then we denote by $G_{(U)}$ the\ngroup of elements $g\\in G$ acting trivially outside of $U$, see\nSubsection~\\ref{ss:somegeneralfacts}.\n\nThe following theorem is proved in~\\cite[Theorem~0.2]{rubin:reconstr}.\n\n\\begin{theorem}\n\\label{th:rubin}\nLet $G_i$, for $i=1, 2$, be groups acting faithfully by homeomorphisms on Hausdorff\ntopological spaces $\\mathcal{X}_i$. Suppose that the following conditions hold\nfor both pairs $(G, \\mathcal{X})=(G_i, \\mathcal{X}_i)$, $i=1, 2$.\n\\begin{enumerate}\n\\item For every non-empty open subset $U\\subset\\mathcal{X}$ the group $G_{(U)}$\n  is non-trivial.\n\\item For every non-empty open subset $U\\subset\\mathcal{X}$ there exists a\n  non-empty open subset $U_1\\subseteq U$ such that if $V, W\\subset\n  U_1$ are open sets such that there exists $g\\in G$ such that\n  $g(V)\\cap W\\ne\\emptyset$, then there exists $g\\in G_{(U)}$ such that\n  $g(V)\\cap W\\ne\\emptyset$.\n\\end{enumerate}\nThen for every isomorphism $\\phi:G_1\\longrightarrow G_2$ there exists a\nhomeomorphism $F:\\mathcal{X}_1\\longrightarrow\\mathcal{X}_2$ inducing it, i.e., such that\n$\\phi(g)=F\\circ g\\circ F^{-1}$ for all $g\\in G$.\n\\end{theorem}\n\n\nWe say that a group $G$ acting on a topological space $\\mathcal{X}$\nis \\emph{locally transitive} if there exists a basis of open sets\n$\\mathcal{U}$ such that for every $U\\in\\mathcal{U}$ the\ngroup $G_{(U)}$ has a dense orbit in $U$.\n\nThe following is a direct corollary of Theorem~\\ref{th:rubin}.\n\n\\begin{corollary}\n\\label{cor:rubin}\nIf $G_i$ are locally transitive groups of homeomorphisms of\ntopological spaces $\\mathcal{X}_i$, then every isomorphism $\\phi:G_1\\longrightarrow G_2$\nis induced by a homeomorphism $F:\\mathcal{X}_1\\longrightarrow\\mathcal{X}_2$.\n\\end{corollary}\n\nSimilar results (with simpler proofs), which can be applied to many\ngroups $\\mathcal{V}_G$, are proved in~\\cite{gior:full,medynets:reconstruction,matui:fullonesided}.\n\nIt is easy to see that the Higman-Thompson group $\\mathcal{V}_{|X|}$ acting on the space\n$X^\\omega$ is locally transitive. It follows that every group of\nhomeomorphisms of $X^\\omega$ containing the Higman-Thompson group is\nlocally transitive, which implies the following fact.\n\n\\begin{theorem}\n\\label{th:vfrubin}\nLet $G_i$ be groups acting on the Cantor sets $X_i^\\omega$ and\ncontaining the Higman-Thompson groups $\\mathcal{V}_{|X_i|}$. Then every\nisomorphism $\\phi:G_1\\longrightarrow G_2$ is induced by a homeomorphism\n$F:X_1^\\omega\\longrightarrow X_2^\\omega$.\n\\end{theorem}\n\n\\subsection{Proof of Theorem~\\ref{th:classification}}\n\nIf $f_1:\\mathcal{J}_1\\longrightarrow\\mathcal{J}_1$ and $f_2:\\mathcal{J}_2\\longrightarrow\\mathcal{J}_2$ are topologically\nconjugate self-coverings of path-connected spaces, then the groups\n$\\mathcal{V}_{f_1}$ and $\\mathcal{V}_{f_2}$ are obviously isomorphic, since they were\ndefined in purely topological terms.\n\nLet us prove the converse implication for expanding maps. \nBy Theorem~\\ref{th:vfrubin},\nif groups $\\mathcal{V}_{f_1}$ and $\\mathcal{V}_{f_2}$ are isomorphic, then their action\non the corresponding spaces $X_i^\\omega$ are topologically conjugate,\nhence the groupoid of germs of the action of $\\mathcal{V}_{f_i}$ on $X_i^\\omega$\nare isomorphic.\n\nTherefore, it is enough to show that if $f:\\mathcal{J}\\longrightarrow\\mathcal{J}$ is an\nexpanding self-covering of a compact path-connected metric space, then\nthe dynamical system $(f, \\mathcal{J})$ can be reconstructed from the\ntopological groupoid $\\mathfrak{G}$ of germs of the action of $\\mathcal{V}_f$ on\n$X^\\omega$.\n\nDenote by $\\mathfrak{F}$ the groupoid of germs generated by\n$f:\\mathcal{J}\\longrightarrow\\mathcal{J}$. We identify $\\mathcal{J}$ with the limit space $\\mathcal{J}_G$ of the\nself-similar group $G=\\img{f}$, and hence encode points of $\\mathcal{J}$ by\nsequences $\\ldots x_2x_1\\in X^{-\\omega}$. Recall that $f$ acts then by\nthe shift $\\ldots x_2x_1\\mapsto\\ldots x_3x_2$. Let\n$\\nu:\\mathfrak{F}\\longrightarrow\\mathbb{Z}$ be the cocycle (groupoid homomorphism)\ndefined by the condition that $\\nu(f, x)=-1$ for all $x\\in\\mathcal{J}$.\n\nThe groupoids $\\mathfrak{F}$ and $\\mathfrak{G}$ are hyperbolic and mutually\ndual. Let $w\\in X^\\omega$ be an arbitrary point, and\ndenote $\\mathfrak{H}=\\mathfrak{d}\\mathfrak{G}_w$ and $\\mathcal{H}=\\partial\\mathfrak{G}_w=\\mathfrak{H}^{(0)}$. It is enough\nto show that $(f, \\mathcal{J})$ is uniquely determined (up to a topological\nconjugacy) by the groupoid $\\mathfrak{H}$.\n\nDenote by $\\Omega_w$ the set of bi-infinite sequences $\\ldots\nx_{-2}x_{-1}.x_0x_1\\ldots$ such that $x_0x_1\\ldots$ belongs to the\n$G$-orbit of $w$. Note that $\\mathcal{J}$ is path connected,\n$G$ is self-replicating, hence $G$-orbits coincide with the\n$\\mathcal{V}_f=\\mathcal{V}_G$-orbits. We consider $\\Omega_w$ with the topology of the disjoint\nunion of the set of the form $X^{-\\omega}.x_0x_1\\ldots$.\n\nThen the space $\\mathcal{H}=\\partial\\mathfrak{G}_w$ is naturally identified with the quotient of the\nspace $\\Omega_w$ by the asymptotic equivalence relation (defined in\nthe same way as on $X^{\\mathbb{Z}}$, see Subsection~\\ref{ss:solenoid}). Let\n$P_S:\\mathcal{H}\\longrightarrow\\mathcal{J}$ be the natural\nprojection induced by $\\ldots x_{-2}x_{-1}.x_0x_1\\ldots\\mapsto\\ldots\nx_{-2}x_{-1}$. It is a covering map, and $\\mathfrak{H}$ is the\nlift of $\\mathfrak{F}$ by $P_S$.\nWe will also denote by $P_S$ the corresponding functor (homomorphism of groupoids)\n$P_S:\\mathfrak{H}\\longrightarrow\\mathfrak{F}$. \n\nLet us show at first that the cocycle $\\nu:\\mathfrak{H}\\longrightarrow\\mathbb{Z}$ (equal to the\nlift of the cocycle $\\nu:\\mathfrak{F}\\longrightarrow\\mathbb{Z}$) is\nuniquely determined by the structure of the topological groupoid\n$\\mathfrak{H}$.\n\n\\begin{proposition}\n\\label{pr:concomp}\nLet $\\mathcal{C}$ be a connected component of $\\mathfrak{H}$. Then\n$\\mathsf{o}:\\mathcal{C}\\longrightarrow\\mathcal{H}$, $\\mathsf{t}:\\mathcal{C}\\longrightarrow\\mathcal{H}$\nare coverings. \n\nIf $\\nu(\\mathcal{C})\\ne 0$, then $\\mathcal{C}$\ncontains a non-trivial element of infinite order in the isotropy group $\\mathfrak{H}_x$\nof a point.\n\nIf $\\nu(\\mathcal{C})=0$, then the groupoid generated by $\\mathcal{C}$\nis proper.\n\\end{proposition}\n\n\\begin{proof}\nLet $\\mathcal{X}_G$ be the limit $G$-space. The action of $G$ on\n$\\mathcal{X}_G$ is free, and the maps $F_v:\\xi\\mapsto\\xi\\otimes v$ are coverings\nfor all $v\\in X^n\\cdot G$.\n\nFor $w\\in X^\\omega$, the leaf $\\partial\\mathfrak{G}_w=\\mathcal{H}$ is the image of $\\mathcal{X}_G$\nunder the map $P_w:\\xi\\mapsto \\xi\\cdot w$.\nThis map coincides with the quotient of $\\mathcal{X}_G$ by the action of the\nstabilizer $G_w$.\n\nLet $\\mathfrak{X}$ be the groupoid of germs with the space of units\n$\\mathcal{X}_G$ generated by the germs of the action of $G$ and the germs of\nthe maps $F_v(\\xi)=\\xi\\otimes v$ for $v\\in X^*\\cdot G$. Then\n$\\mathfrak{X}$ is the lift\nof $\\mathfrak{H}$ by the quotient map $P_w:\\mathcal{X}_G\\longrightarrow\\mathcal{H}$.\n\nEvery element of $\\mathfrak{H}$ is a germ of the transformation \n\\[F_{v\\cdot g, u\\cdot h}:\\xi\\otimes v.g(w)\\mapsto\\xi\\otimes u.h(w),\\]\nfor some $g, h\\in G$ and $u, v\\in X^*$. \n\nThe germ $(F_{v\\cdot g, u\\cdot h}, \\zeta\\otimes v.g(w))$\ncan be lifted to the germ $(\\tilde F_{v\\cdot g, u\\cdot h},\n\\zeta\\otimes v\\cdot g)$ of\nthe local homeomorphism\n\\[\\tilde F_{v\\cdot g, u\\cdot h}:\\xi\\otimes v\\cdot g\\mapsto\\xi\\otimes u\\cdot h\\]\nof $\\mathfrak{X}$. It follows that every element of $\\mathfrak{H}$ is\na germ of $P_wF_{u\\cdot h}F_{v\\cdot g}^{-1}P_w^{-1}$. The\nspace $\\mathcal{X}_G$ is connected, the maps $F_{u\\cdot h}, F_{v\\cdot g},\nP_w$ are coverings, hence if $\\mathcal{C}$ is the connected component of\nthe germ $(F_{v\\cdot g, u\\cdot h}, \\zeta\\otimes v.g(w))$, then\n$\\mathsf{o}:\\mathcal{C}\\longrightarrow\\mathcal{H}$ and $\\mathsf{t}:\\mathcal{C}\\longrightarrow\\mathcal{H}$ are covering maps.\n\nSuppose that $\\nu(\\mathcal{C})\\ne 0$. It means that every germ\n$(F_{v\\cdot g, u\\cdot h}, \\zeta\\otimes v.g(w))\\in\\mathcal{C}$ is such that $|v|\\ne\n|u|$. Without loss of generality, we may assume that $|u|>|v|$. Let\n$u=u_1v_1$, where $|v_1|=|v|$. Since $G$ is self-replicating, there\nexists $g_1\\in G$ such that $g_1\\cdot v\\cdot g=v_1\\cdot h$ in the\nbiset. Then a lift of the germ $(F_{v\\cdot g, u\\cdot h}, \\zeta\\otimes\nv.g(w))$ to $\\mathfrak{X}$ is a germ of the\ntransformation\n\\[\\xi\\otimes v\\cdot\ng\\mapsto \\xi\\otimes u_1\\otimes v_1\\cdot h=\\xi\\otimes u_1\\cdot\ng_1\\otimes v\\cdot g.\\] The point $\\zeta=\\ldots u_1\\cdot g_1\\otimes u_1\\cdot\ng_1\\otimes u_1\\cdot g_1\\in\\mathcal{X}_G$ is well defined (as it is the image of\na point of $\\Omega$, see Subsection~\\ref{ss:solenoid}), and it satisfies\n$\\zeta=\\zeta\\otimes u_1\\cdot g_1$. Then the germ of the transformation\n\\[\\xi\\otimes v\\cdot g\\mapsto \\xi\\otimes u_1\\cdot g\\otimes v\\cdot g\\]\nat $\\zeta\\otimes v\\cdot g$ is a non-trivial\ncontracting element of the isotropy group of $\\zeta\\otimes v\\cdot\ng$. It is contained in the connected component of the germs of the\ntransformation\n$\\xi\\otimes v\\cdot g\\mapsto \\xi\\otimes u\\cdot h$. Mapping everything to\n$\\mathfrak{H}$ by $P_w$, we find a non-trivial contracting (hence infinite\norder) element of an isotropy group.\n\nIf $\\nu(\\mathcal{C})=0$, then elements of $\\mathcal{C}$ are germs of\ntransformations of the form $\\xi\\otimes\nv.g(w)\\mapsto\\xi\\otimes u.h(w)$, where $g, h\\in G$ and $v, u\\in X^*$\nare such that $|v|=|u|$. There exists $g_1\\in G$ such that $u\\cdot\nh=g_1\\cdot v\\cdot g$ in $X^n\\cdot G$. It follows that elements of\n$\\mathcal{C}$ are lifted by $P_wF_{v\\cdot g}:\\mathfrak{X}\\longrightarrow\\mathfrak{H}$ to the\naction of $g_1$ on $\\mathcal{X}_G$. It follows that the groupoid generated by\n$\\mathcal{C}$ lifts by $P_wF_{v\\cdot g}$ to a subgroupoid of the action of $G$ on $\\mathcal{X}_G$,\nand hence is proper.\n\\end{proof}\n\n\\begin{proposition}\n\\label{pr:cocycleunique}\nThe cocycle $\\nu:\\mathfrak{H}\\longrightarrow\\mathbb{Z}$ is uniquely determined by the topological\ngroupoid $\\mathfrak{H}$.\n\\end{proposition}\n\n\\begin{proof}\nIt follows from~\\ref{pr:concomp} that \nthe value of $\\nu$ on a connected component\n$\\mathcal{C}$ of $\\mathfrak{H}$ is zero if and only if $\\mathcal{C}$\ngenerates a proper groupoid. (Since isotropy groups of a proper\ngroupoid are finite.)\n\nLet $g_1, g_2$ be arbitrary elements of $\\mathfrak{H}$. By the first claim of\nProposition~\\ref{pr:concomp}, there exist $g_1', g_2'$ in the components of $g_1$\nand $g_2$, respectively, such that the product $g_1'g_2'$ is\ndefined. Note that the connected component of $g_1'g_2'$ depends only\non the connected components of $g_1$ and $g_2$. It follows that the\nset of connected components of $\\mathfrak{H}$ is a group. The\nquotient of this group by the subgroup of components on which $\\nu$\nis zero is isomorphic to $\\mathbb{Z}$. Since the set of components on which\n$\\nu$ is zero is uniquely determined by the topological groupoid, we\nconclude that the set $\\{\\nu, -\\nu\\}$ is uniquely determined by the\nstructure of the topological groupoid $\\mathfrak{H}$. But we can distinguish\nbetween $\\nu$ and $-\\nu$ using~\\cite[Proposition~3.4.1.]{nek:hyperbolic}.\n\\end{proof}\n\nThe next statement follows now directly from\nProposition~\\ref{pr:cocycleunique} and Corollary~\\ref{cor:solenoidflow}.\n\n\\begin{proposition}\nThe natural extension $\\hat f:\\hat\\mathcal{J}\\longrightarrow\\hat\\mathcal{J}$ is uniquely\ndetermined, up to topological conjugacy, by the groupoid $\\mathfrak{G}$.\n\\end{proposition}\n\nSuppose that $f_1:\\mathcal{J}_1\\longrightarrow\\mathcal{J}_1$ and $f_2:\\mathcal{J}_2\\longrightarrow\\mathcal{J}_2$ are two\nexpanding homeomorphisms with the same natural extension\n$\\hat f:\\mathcal{S}\\longrightarrow\\mathcal{S}$. It remains to prove that $(f_1, \\mathcal{J}_1)$\nand $(f_2, \\mathcal{J}_2)$ are topologically conjugate.\n\nDenote by \n$P_i:\\mathcal{S}\\longrightarrow\\mathcal{J}_i$ the corresponding projections.\nLet $\\tilde\\mathcal{J}$ be the image of $\\mathcal{S}$ in $\\mathcal{J}_1\\times\\mathcal{J}_2$ under the\nmap $(P_1, P_2)$. It is compact and connected, since so is\n$\\mathcal{S}$. We will denote by\n$\\tilde P_i:\\tilde\\mathcal{J}\\longrightarrow\\mathcal{J}_i$ the restrictions of the projections\n$\\mathcal{J}_1\\times\\mathcal{J}_2\\longrightarrow\\mathcal{J}_i$.\n\nSince $P_i$ locally are projections on the unstable coordinate of\nthe local product decomposition of $\\mathcal{S}$ (which depends only\non the conjugacy class of $(\\hat f, \\mathcal{S})$), for every\n$\\xi\\in\\mathcal{S}$ there exists a rectangular neighborhood $U\\ni\\xi$\nsuch that $P_i:U\\longrightarrow\\pi_i(U)$ is decomposed into the composition of\nprojection of $U$ onto its unstable direction and a homeomorphism of\nthis direction with $P_i(U)$. Moreover, since $\\mathcal{S}$ is\ncompact, we can cover $\\mathcal{S}$ by a finite number of such\nrectangles $U$. \n\nThe map $\\hat f:\\mathcal{S}\\longrightarrow\\mathcal{S}$ induces a map\n$\\tilde f:\\tilde\\mathcal{J}\\longrightarrow\\tilde\\mathcal{J}$ by the rule $\\tilde f(\\xi_1, \\xi_2)=(f_1(\\xi_1),\nf_2(\\xi_2))$. The projections $\\tilde P_i$ are semi-conjugacies of $\\tilde f$ with\n$f_i$.\n\nLet $(\\xi_1, \\xi_2)\\in\\tilde\\mathcal{J}$, i.e., there exists\n$\\xi\\in\\mathcal{S}$ such that $\\xi_i=P_i(\\xi)$. There exists a\nrectangular neighborhood $U$ of $\\xi$ such that $P_i$ are\nprojections onto the unstable direction composed with a\nhomeomorphism, and the unstable direction of $U$ is connected.\nIf $U$ is small enough, then $(\\hat f)^{-1}(U)$ is decomposed\ninto a union of a finite set $\\mathcal{R}$ of rectangles on which each of $P_i$\nis a homeomorphism with projection onto the unstable direction.\nConsider the sets $(P_1, P_2)(R)$ for $R\\in\\mathcal{R}$.\nWe get a finite number of components of $(\\tilde\nf)^{-1}((P_1, P_2)(U))$ such that $\\tilde f$ is a homeomorphism on\neach of them. It follows that $\\tilde f$ is a finite degree covering map.\n\nFor every $\\xi\\in\\mathcal{J}_1$ the set $P_1^{-1}(\\xi)$ is a compact subset of\n$\\mathcal{S}$ contained in one stable equivalence class. Consequently,\nthere exists $n_0$ such that $P_2(\\hat f^n(P_1^{-1}(\\xi)))$ is a\nsingle point for all $n\\ge n_0$. It follows that there exists a small\nneighborhood $U$ of $\\xi$ such that the map $P_2\\circ \\hat f^{n_0}\\circ\nP_1^{-1}=f_2^{n_0}\\circ P_2\\circ P_1^{-1}$ is a homeomorphism on $U$.\nBy compactness, there exists\n$n_1$ such that $P_2\\circ f^{n_1}\\circ P_1^{-1}=f_2^{n_1}\\circ \nP_2\\circ P_1^{-1}$ is a well defined covering map from $\\mathcal{J}_1$ to $\\mathcal{J}_2$.\n\nIt follows that the projections $\\tilde P_i:\\tilde\\mathcal{J}\\longrightarrow\\mathcal{J}_i$ are finite degree\ncovering maps.  For every point $t^{(j)}_i\\in\n{\\tilde P}_i^{-1}(t_i)$ we have the corresponding tree $T_{t^{(j)}_i}$\nof preimages under iterations of $\\tilde f$. They\nare disjoint (more precisely, for every $n$ the sets ${\\tilde f}^{-n}(t^{(j)})$\nare disjoint for different $t^{(j)}$).\n\nBy the arguments above, there exists\n$n_1$  such that $\\tilde P_i(z_1)=\\tilde P_i(z_2)$ implies\n${\\tilde f}^{n_1}(z_1)={\\tilde f}^{n_1}(z_2)$.\nBut this contradicts the fact that the trees\n$T_j$ are disjoint. It follows that $\\tilde P_i$ have degree 1, i.e., are\nhomeomorphisms conjugating $f$ with $f_i$.\n\n\\subsection{Equivalence of groupoids}\n\n\\begin{theorem}\n\\label{th:groupoidequivalence}\nLet $f_i:\\mathcal{J}_i\\longrightarrow\\mathcal{J}_i$, for $i=1, 2$, be expanding self-coverings of\nconnected and locally connected compact metric spaces. Then the\nfollowing conditions are equivalent.\n\\begin{enumerate}\n\\item The dynamical systems $(f_1, \\mathcal{J}_1)$ and $(f_2, \\mathcal{J}_2)$ are\n  topologically conjugate.\n\\item The groupoids generated by germs of $f_1$ and $f_2$ are equivalent.\n\\item The natural extensions of $f_1$ and $f_2$ are topologically\n  conjugate.\n\\item The natural extensions of $f_1$ and $f_2$ generate equivalent\n  groupoids of germs.\n\\item The actions of $\\mathcal{V}_{f_1}$ and $\\mathcal{V}_{f_2}$ on the corresponding\n  Cantor sets are topologically conjugate.\n\\item The groupoids of germs generated by the actions of $\\mathcal{V}_{f_1}$\n  and $\\mathcal{V}_{f_2}$ on the corresponding Cantor sets are equivalent.\n\\item The self-similar groups $\\img{f_1}$ and $\\img{f_2}$ are\n  equivalent.\n\\item The groups $\\mathcal{V}_{f_1}$ and $\\mathcal{V}_{f_2}$ are isomorphic as abstract\n  groups.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nThe groupoid of germs $\\mathfrak{F}_i$ generated by $f_i$, the groupoid of germs\n$\\mathfrak{G}_i$ generated by $\\mathcal{V}_{f_i}$, and the groupoid of germs generated by the\nnatural extension uniquely determine each other, up to equivalence of\ngroupoids, since the first two are mutually dual hyperbolic groupoids,\nand the third one is their geodesic flow, see~\\cite{nek:hyperbolic}.\nEquivalence of (7) and (1) is proved in~\\cite{nek:book}.\n\nIt remains, therefore, to prove that the equivalence class of\n$\\mathfrak{F}_i$ uniquely determines $(f_i, \\mathcal{J}_i)$.\n\nSuppose that $w_1$ and $w_2$ belong to one orbit of the groupoid $\\mathfrak{G}$\nfrom Definition~\\ref{def:equivalentgroupoids}.\nLet $g\\in\\mathfrak{G}$ be such that $\\mathsf{o}(g)=w_2$ and\n$\\mathsf{t}(g)=w_1$. Then the map $h\\mapsto hg$ is a quasi-isometry between the\nCayley graphs of $\\mathfrak{G}_1$ and $\\mathfrak{G}_2$ based at $w_1$ and $w_2$ respectively,\ninducing an isomorphism $\\mathfrak{d}\\mathfrak{G}_{w_1}\\longrightarrow\\mathfrak{d}\\mathfrak{G}_{w_2}$. We have\nshown during the proof of Theorem~\\ref{th:classification}\nthat the dynamical systems $(f_i, \\mathcal{J}_i)$ can be uniquely\nreconstructed from the topological groupoids $\\mathfrak{d}\\mathfrak{G}_{w_i}$, which\nimplies that $(f_1, \\mathcal{J}_1)$ and $(f_2, \\mathcal{J}_2)$ are topologically conjugate.\n\\end{proof}\n\n\\def$'${$'$}\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n  \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}