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+{"text":"\\section{Lower Bound}\n\nTo prove the lower bound on the round complexity of approximating (weighted) diameter in the quantum CONGEST model, we combine the reduction in~\\cite{Elkin06,SarmaHKKNPPW12,ElkinKNP14} and the graph gadget in~\\cite{AbboudCK16}.\n\n\n\n\\subsection{Reduction from Server Model}\n\\label{sec:reduction}\n\nWe briefly outline the reduction introduced by Elkin et al.~\\cite{Elkin06,SarmaHKKNPPW12,ElkinKNP14} from the Server model to prove the hardness of certain graph problems such as diameter and radius.\nWe will introduce a distributed network $G=(V,E)$ and embed a certain two-argument function $F:\\{0,1\\}^k\\times\\{0,1\\}^k\\to\\{0,1\\}$ into the network by showing that if the instance on the network $G$ has a low round-complexity protocol in the quantum CONGEST model, then there exists a low communication complexity protocol for $F$ in the quantum Server model.\nThus, the hardness of diameters and radius in the quantum CONGEST model is reduced to proving the lower bounds the communication complexity in the quantum Server model.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.5\\linewidth]{figures\/example.pdf}\n\\caption{An example of constructed graph $G$.}\n\\label{fig:example}\n\\end{figure}\n\nThe network $G=(V,E)$ is depicted by Figure~\\ref{fig:example} where $V=V_S\\uplus V_A\\uplus V_B$ and $E=E_S\\uplus E_A\\uplus E_B\\uplus E'$.\nWe use $G[U]$ to denote the subgraph induced by vertex set $U\\subseteq V$, then $E_S,E_A,E_B$ are the edges in $G[V_S],G[V_A],G[V_B]$ respectively.\nAnd $E'$ denotes the edges between $V_S$ and $V_A\\uplus V_B$.\n\n$G[V_S]$ includes a full binary tree of height $h$ and $m$ disjoint paths of length $2^h-1$.\nEach of the $2^h$ leaves of the binary tree is connected to the nodes on the paths as depicted in Figure~\\ref{fig:example}.\nSuppose nodes of depth $i$ on the tree are $t_{i,1},\\cdots,t_{i,2^i}$ and nodes on the $i$-th path are $p_{i,1},\\ldots,p_{i,2^h}$ from left to right.\nThen $t_{h,1},\\ldots, t_{h,2^h}$ are the leaves of the binary tree in $G[V_S]$.\nFor each $i\\in[1,m]$ and $j\\in[1,2^h]$, there is an edge between $t_{h,j}$ and $p_{i,j}$.\nThus,\n$$\n\\begin{aligned}\nV_S & =\\left\\{t_{i,j}:i\\in[0,h],j\\in[1,2^i]\\right\\} \\\\\n& \\uplus\\left\\{p_{i,j}:i\\in[1,m],j\\in[1,2^i]\\right\\}, \\\\\nE_S & =\\left\\{\\{t_{i,j},t_{i-1,\\lceil j\/2\\rceil}\\}:i\\in[1,h],j\\in[1,2^i]\\right\\} \\\\\n& \\uplus\\left\\{\\{p_{i,j},p_{i,j-1}\\}:i\\in[1,m],j\\in[2,2^m]\\right\\} \\\\\n& \\uplus\\left\\{\\{t_{h,j},p_{i,j}\\}:i\\in[1,m],j\\in[1,2^m]\\right\\}.\n\\end{aligned}\n$$\n$V_A$ contains at least $m$ nodes, each of which is connected to $p_{i,1}$ for $1\\leq i\\leq m$.\n$V_B$ contains at least $m$ nodes, each of which is connected to $p_{i,2^h}$ for $1\\leq i\\leq m$.\nThose $2m$ edges are contained in $E'$.\nThe subgraphs $G[V_A]$ and $G[V_B]$ are decided by Alice's input and Bob's input, respectively.\n\n\\bigskip\n\nThe following lemma gives an efficient simulation of algorithms on network $G$ by the protocols in the quantum Server model.\n\n\\begin{lemma}[Quantum Simulation Lemma]\nSuppose Alice and Bob are given $(V_A,E_A)$ and $(V_B,E_B)$, respectively.\nFor any $T$-round ($T<2^h\/2$) distributed algorithm on network $G$ described above, there exists a communication protocol for Alice and Bob in the quantum Server model to simulate the algorithm with communication complexity $O(T\\cdot h\\cdot B)$,  where $B$ denotes the bandwidth in the CONGEST model.\n\\label{lem:simulation}\n\\end{lemma}\n\n\\begin{proof}\nThe proof of Lemma~\\ref{lem:simulation} follows closely with the proof in \\cite[Proof of Theorem 3.5]{ElkinKNP14}.\nThe protocol we will construct simulates the distributed algorithm round by round.\nThus, it also has $T<2^h\/2$ rounds of communication.\nIn the beginning, the server simulates all the nodes in $V_S$ which are independent of Alice and Bob's inputs.\nAnd in the end of the $r$-th round, the server simulates $p_{i,1+r},\\cdots,p_{i,2^h-r}$ on the $i$-th path and nodes $t_{h,1+r},\\cdots,t_{h,2^h-r}$ along with their ancestors on the binary tree, while Alice simulates the nodes on the left side and Bob simulates on the right side.\nMore formally, in the end of the $r$-th round, the server simulates\n$$\\left\\{p_{i,j}:i\\in[1,m],j\\in[1+r,2^h-r]\\right\\}\\cup\\left\\{t_{i,j}:i\\in[0,h],j\\in\\left[\\left\\lceil(1+r)\/2^{h-i}\\right\\rceil,\\left\\lceil(2^h-r)\/2^{h-i}\\right\\rceil\\right]\\right\\};$$\nAlice simulates\n$$V_A\\cup\\left\\{p_{i,j}:i\\in[1,m],j\\in[1,1+r)\\right\\}\\cup\\left\\{t_{i,j}:i\\in[0,h],j\\in\\left[1,\\left\\lceil(1+r)\/2^{h-i}\\right\\rceil\\right)\\right\\};$$\nBob simulates\n$$V_B\\cup\\left\\{p_{i,j}:i\\in[1,m],j\\in(2^h-r,2^h]\\right\\}\\cup\\left\\{t_{i,j}:i\\in[0,h],j\\in\\left(\\left\\lceil(2^h-r)\/2^{h-i}\\right\\rceil,2^i\\right]\\right\\}.$$\n\nWe describe the simulation of the computation and communication of a processor $v$ in the $r$-th round, and count the total communication complexity.\n\\begin{itemize}\n\\item If $v$ is owned by Alice or the server in the $(r-1)$-th round and will be owned by Alice in the $r$-th round, Alice needs the local information of $v$ in the $(r-1)$-th round and messages from $\\Gamma(v)$ (neighbours of $v$) to $v$ in the $r$-th round, which can be obtained by local computation and communication from the server to Alice since $T<2^h\/2$, which implies that each of $v$ and nodes in $\\Gamma(v)$ is owned by either Alice or the server in the $(r-1)$-th round for $r\\leq T$.\nSo in this case, we only need communication from the server to Alice in the Server model.\nThis part will not be counted to complexity by definition.\n\\item If $v$ is owned by Bob or the server in the $(r-1)$-th round and will be owned by Bob in the $r$-th round, no communication will be counted to complexity by the same argument as mentioned above.\n\\item If $v$ is owned by the server in both the $(r-1)$-th round and the $r$-th round,\nthe server needs the messages from $\\Gamma(v)$ to $v$.\nFor each node $u\\in\\Gamma(v)$ owned by Alice or Bob in the $(r-1)$-th round, Alice or Bob will simulates the local computation of $u$ in the $r$-th round, and send the message to the server.\n\\begin{itemize}\n\\item If $v$ is on the paths on $V_S$, none of $\\Gamma(v)$ is owned by Alice and Bob in the $(r-1)$-th round.\n\\item If $v$ is on the binary tree, node $u\\in\\Gamma(v)$ is owned by Alice in the $(r-1)$-th round only if all nodes of the same depth with $v$, meanwhile on the left side of $v$, are not owned by the server in the $r$-th round, and $u$ is the left-child of $v$.\nSimilarly, node $u\\in\\Gamma(v)$ is owned by Bob in the $(r-1)$-th round only if all nodes of the same depth with $v$, meanwhile on the right side of $v$, are not owned by the server in the $r$-th round, and $u$ is the right-child of $v$.\nIn the $r$-th round, there are at most $2h$ such $(u,v)$ in total.\n\\end{itemize}\n\\end{itemize}\nHence, a total of $O(T\\cdot h)$ messages, each of size $O(B)$, are sent from Alice or Bob to the server.\n\\end{proof}\n\n\n\n\\subsection{Hardness of Approximating Diameter}\n\nwe will use $G$ constructed above as a gadget to prove a lower bound on round complexity of approximating weighted diameter in the quantum CONGEST model.\nThe specific graph depicted in Figure~\\ref{fig:diameter} will contain $n=\\left(2^{h+1}-1\\right)+\\left(2s+\\ell\\right)\\left(2^h+2\\right)+2\\cdot2^s$ nodes, where parameters $h,s,\\ell$ are chosen as follows throughout this section.\n\\begin{equation}\n\\label{eqn:paramters}\nh\\text{ is some even number},s=3h\/2,\\ell=2^{s-h}.\n\\end{equation}\nThis choice makes $2^h=\\widetilde\\Theta(n^{2\/3}),2^s=\\widetilde\\Theta(n)$ and $\\ell=\\widetilde\\Theta(n^{1\/3})$.\n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=\\linewidth]{figures\/Diameter.pdf}\n\\caption{\nGraph $G$ for approximating diameter.\nThe black edges are of weight $1$; the blue edges are of weight $\\alpha$; and weights of red edges are determined by inputs $x,y$, i.e, $w(\\{a_i,a^\\star_j\\})=\\alpha$ if $x_{i,j}=1$ and $w(\\{a_i,a^\\star_j\\})=\\beta$ if $x_{i,j}=0$, and $w(\\{b_i,b^\\star_j\\})=\\alpha$ if $y_{i,j}=1$ and $w(\\{b_i,b^\\star_j\\})=\\beta$ if $y_{i,j}=0$, for $i\\in[1,2^s]$ and $j\\in[1,\\ell]$.\n}\n\\label{fig:diameter}\n\\end{figure*}\n\n\\begin{theorem}[Restated]\nFor any constant $\\varepsilon\\in(0,\\frac12]$, any algorithm, with probability at least $\\frac{11}{12}$, computing a $(\\frac32-\\varepsilon)$-approximation of the weighted diameter in the quantum CONGEST model requires $\\Omega\\left(\\frac{n^{2\/3}}{\\log^2n}\\right)$ rounds, even when the unweighted diameter is $\\Theta(\\log n)$, where $n$ denotes the number of nodes.\n\\label{thm:diameter}\n\\end{theorem}\n\nOn network $G=(V,E)$ described in Section~\\ref{sec:reduction}, we specify $G[V_A]$ and $G[V_B]$.\nLet\n\\begin{equation*}\n\\begin{aligned}\nV_A & =\\left\\{a_1,\\cdots,a_{2^s}\\right\\}\\uplus\\left\\{a^0_1,a^1_1,\\cdots,a^0_s,a^1_s\\right\\}\\uplus\\left\\{a^\\star_1,\\cdots,a^\\star_\\ell\\right\\}, \\\\%\\text{ and} \\\\\nV_B & =\\left\\{b_1,\\cdots,b_{2^s}\\right\\}\\uplus\\left\\{b^0_1,b^1_1,\\cdots,b^0_s,b^1_s\\right\\}\\uplus\\left\\{b^\\star_1,\\cdots,b^\\star_\\ell\\right\\}.\n\\end{aligned}\n\\end{equation*}\nThe edges $E_A, E_B$ and $E'$ are specified as follows.\n$$\n\\begin{aligned}\nE_A & =\\left\\{\\{a_i,a^{\\text{bin}(i,j)}_j\\}:i\\in[1,2^s],j\\in[1,s]\\right\\} \\\\\n& \\uplus\\left\\{\\{a_i,a^\\star_j\\}:i\\in[1,2^s],j\\in[1,\\ell]\\right\\} \\\\\n& \\uplus\\left\\{\\{a_i,a_j\\}:i,j\\in[1,2^s],i\\ne j\\right\\}, \\\\\nE_B & =\\left\\{\\{b_i,b^{\\text{bin}(i,j)}_j\\}:i\\in[1,2^s],j\\in[1,s]\\right\\} \\\\\n& \\uplus\\left\\{\\{b_i,b^\\star_j\\}:i\\in[1,2^s],j\\in[1,\\ell]\\right\\} \\\\\n& \\uplus\\left\\{\\{b_i,b_j\\}:i,j\\in[1,2^s],i\\ne j\\right\\}, \\\\\nE' & =\\left\\{\\{a^0_i,p_{2i-1,1}\\},\\{b^1_i,p_{2i-1,2^h}\\}:i\\in[1,s]\\right\\} \\\\\n& \\uplus\\left\\{\\{a^1_i,p_{2i,1}\\},\\{b^0_i,p_{2i,2^h}\\}:i\\in[1,s]\\right\\} \\\\\n& \\uplus\\left\\{\\{a^\\star_i,p_{2s+i,1}\\},\\{b^\\star_i,p_{2s+i,2^h}\\}:i\\in[1,\\ell]\\right\\},\n\\end{aligned}\n$$\nwhere  $\\text{bin}(i,j)$ denote the $j$-th bit in binary expression of integer $i-1$.\n\nThe node pairs $(a^0_i, p_{2i-1,1})$, $(a^1_i,p_{2i,1})$, $(b^0_i,p_{2i,2^h})$, $(b^1_i,p_{2i-1,2^h})$ for $1\\le i\\le s$, and $(a^\\star_j,p_{2s+j,1})$, $(b^\\star_j,p_{2s+j,2^h})$ for $1\\le j\\le\\ell$ are connected.\nFor each $i\\in[1,2^s]$, $a_i$ is connected to $a^{\\text{bin}(i,j)}_j$ for each $j\\in[1,s]$, and $a_i$ is connected to $a^\\star_j$ for each $j\\in[1,\\ell]$.\nMoreover, $G[\\{a_1,\\cdots,a_{2^s}\\}]$ is a clique.\nThe edges in $G[V_B]$ are linked in the same way as the edges in $G[V_A]$.\n\nThe weights of the edges are specified as follows, which are also depicted in Figure~\\ref{fig:diameter}.\n\\begin{itemize}\n\\item The edges on the binary tree and the edges on the $2s+\\ell$ paths (including the endpoints in $V_A$ and $V_B$) are of weight $1$ (the black edges in Figure~\\ref{fig:diameter}).\n\\item Recall that Alice and Bob receive inputs $x,y\\in\\{0,1\\}^{2^s\\cdot\\ell}$ respectively.\n$x$ and $y$ are indexed by $x_{i,j}$ and $y_{i,j}$ for $i\\in[1,2^s],j\\in[1,\\ell]$ where $s$ and $\\ell$ are given in Eq.~\\eqref{eqn:paramters}.\nFor each $i\\in[1,2^s],j\\in[1,\\ell]$, $w(\\{a_i,a^\\star_j\\})=\\alpha$ if $x_{i,j}=1$ and $w(\\{a_i,a^\\star_j\\})=\\beta$ if $x_{i,j}=0$ ($\\alpha<\\beta$); weights of edges between $\\{b_1,\\cdots,b_{2^s}\\}$ and $\\{b^\\star_1,\\cdots,b^\\star_\\ell\\}$ are assigned according to $y$ in the same way (the red edges in Figure~\\ref{fig:diameter}).\n\\item The edges between the binary tree and the $2s+\\ell$ paths, those between $\\{a_1,\\cdots,a_{2^s}\\}$ and $\\{a^0_1,a^1_1,\\cdots,a^0_s,a^1_s\\}$, and those between $\\{b_1,\\cdots,b_{2^s}\\}$ and $\\{b^0_1,b^1_1,\\cdots,b^0_s,b^1_s\\}$ are of weight $\\alpha$; weights of edges inside $G[\\{a_1,\\cdots,a_{2^s}\\}]$ and $G[\\{b_1,\\cdots,b_{2^s}\\}]$ are also $\\alpha$ (the blue edges in Figure~\\ref{fig:diameter}).\n\\end{itemize}\n\nIt is sufficient to analyze the diameter of graph after contracting all edges of weight $1$ due to the following lemma.\nAn edge is contracted if the two endpoints are merged to one node, and the adjacent edges of the two endpoints are incident to it.\nIf there are parallel edges after contraction,  we only keep the one with the lowest weight.\n\n\\begin{lemma}\nGiven a weighted graph $(G,w)$ where $G=(V,E)$ and $w:E\\to\\mathbb N^+$.\nLet $G'$ be the graph after contracting all edges of weight $1$.\nWe have $D_{G',w}\\le D_{G,w}\\le D_{G',w}+n$ and $R_{G',w}\\le R_{G,w}\\le R_{G',w}+n$, where $n=|V|$.\n\\label{lem:contraction}\n\\end{lemma}\n\\begin{proof}\nFor any path $P$ in $G$, let $P'$ be the path in $G'$ obtained from $P$ after contraction.\nThen\n$$\\text{length}(P')\\leq\\text{length}(P)\\leq\\text{length}(P')+n$$\nas there are at most $n-1$ $1$-weight edges.\nThus we conclude the result.\n\\end{proof}\n\nFor inputs $x,y\\in\\{0,1\\}^{2^s\\cdot\\ell}$ received by Alice and Bob, define\n$$F(x,y)=\\bigwedge_{i\\in[1,2^s]}\\left(\\bigvee_{j\\in[1,\\ell]}\\left(x_{i,j}\\wedge y_{i,j}\\right)\\right),$$\ni.e., $F=\\text{AND}_{2^s}\\circ(\\text{OR}_\\ell\\circ\\text{AND}^\\ell_2)^{2^s}$.\nWe have the following lemma.\n\n\\begin{lemma}\n$D_{G,w}\\le\\max\\{2\\alpha,\\beta\\}+n$ if $F(x,y)=1$, and $D_{G,w}\\ge\\min\\{\\alpha+\\beta,3\\alpha\\}$ otherwise.\n\\label{lem:reduction}\n\\end{lemma}\n\n\\begin{proof}\nThe graph $G'$ after contraction is given in Figure~\\ref{fig:contraction}.\nThe binary tree is contracted to node $t$.\nThe $2s+\\ell$ paths are contracted to nodes $a^0_1,a^1_1,\\cdots,a^0_s,a^1_s$ and $a^\\star_1,\\cdots,a^\\star_\\ell$ respectively.\nNote that $b_i$ is connected to $a^{\\text{bin}(i,j)\\oplus1}_j$ for $i\\in[1,2^s],j\\in[1,s]$.\nwe list upper bounds of the distances between any two nodes $u$ and $v$ in $G'$ on Table~\\ref{tab:distance} with the corresponding paths, except for the distance between $a_i$ and $b_i$ with $i\\in[1,2^s]$.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.5\\linewidth]{figures\/Contraction.pdf}\n\\caption{\nGraph $G'$ after contraction.\nThe distance between any pair of nodes, except $a_i$ and $b_i$ for $i\\in[1,2^s]$, is at most $\\max\\{2\\alpha,\\beta\\}$; and the distance between $a_i$ and $b_i$ is at most $2\\alpha$ if there exists $j\\in[1,\\ell]$ such that $x_{i,j}=y_{i,j}=1$, otherwise it is at least $\\min\\{\\alpha+\\beta,3\\alpha\\}$.\nTherefore, the diameter is at most $\\max\\{2\\alpha,\\beta\\}$ if, for any $i\\in[1,2^s]$, there exists $j\\in[1,\\ell]$ such that $x_{i,j}=y_{i,j}=1$, otherwise it is at least $\\min\\{\\alpha+\\beta,3\\alpha\\}$.\n}\n\\label{fig:contraction}\n\\end{figure}\n\n\\begin{table*}[t]\n\\centering\n\\caption{\nDistance between nodes in $G'$.\nLet {\\rm router} be any node in $\\{a^0_1,a^1_1,\\cdots,a^0_s,a^1_s,a^\\star_1,\\cdots,a^\\star_\\ell\\}$.\n${\\rm adj}(i,j)$ denotes the integer after changing the $j$-th bit in binary expression of integer $i-1$, and ${\\rm ind}(i,j)$ is the smallest $z\\in[1,s]$ satisfying ${\\rm bin}(i,z)\\ne{\\rm bin}(j,z)$.\n}\n\\label{tab:distance}\n\\begin{tabular}{cccc}\n\\toprule[1.5pt]\n$u$                                         & $v$                                               & $d_{G',w}(u,v)$   & Path \\\\\n\\midrule[1.5pt]\n\\multirow{3.6}{*}{$t$} \t\t\t\t\t\t& router                                            & $\\le\\alpha$       & $\\left(t\\to v\\right)$ \\\\\n\t\t\t\t\t\t\t\t\t\t\t& $a_i$ ($i\\in[1,2^s]$)                             & $\\le2\\alpha$      & $\\left(t\\to a^{\\text{bin}(i,0)}_0\\to a_i\\right)$ \\\\\n\t\t\t\t\t\t\t\t\t\t\t& $b_i$ ($i\\in[1,2^s]$)                             & $\\le2\\alpha$      & $\\left(t\\to a^{\\text{bin}(i,0)\\oplus1}_0\\to b_i\\right)$ \\\\\n\\midrule\n\\multirow{6.4}{*}{$a_i$ ($i\\in[1,2^s]$)}    & $a_j$ ($j\\ne i,j\\in[1,2^s]$)                      & $\\le\\alpha$       & $\\left(a_i\\to a_j\\right)$ \\\\\n\t\t\t\t\t\t\t\t\t\t\t& $a^{\\text{bin}(i,j)}_j$ ($j\\in[1,s]$)             & $\\le\\alpha$       & $\\left(a_i\\to a^{\\text{bin}(i,j)}_j\\right)$ \\\\\n\t\t\t\t\t\t\t\t\t\t\t& $a^{\\text{bin}(i,j)\\oplus1}_j$ ($j\\in[1,s]$)      & $\\le2\\alpha$      & $\\left(a_i\\to a_{\\text{adj}(i,j)}\\to a^{\\text{bin}(i,j)\\oplus1}_j\\right)$ \\\\\n\t\t\t\t\t\t\t\t\t\t\t& $b_j$ ($j\\ne i,j\\in[1,2^s]$)                      & $\\le2\\alpha$      & $\\left(a_i\\to a^{\\text{bin}(i,\\text{ind}(i,j))}_{\\text{ind}(i,j)}\\to b_j\\right)$ \\\\\n\t\t\t\t\t\t\t\t\t\t\t& $a^\\star_j$ ($j\\in[1,\\ell]$)                      & $\\le\\beta$        & $\\left(a_i\\to a^\\star_j\\right)$ \\\\\n\\midrule\n\\multirow{5.2}{*}{$b_i$ ($i\\in[1,2^s]$)}    & $b_j$ ($j\\ne i,j\\in[1,2^s]$)                      & $\\le\\alpha$       & $\\left(b_i\\to b_j\\right)$ \\\\\n\t\t\t\t\t\t\t\t\t\t\t& $a^{\\text{bin}(i,j)\\oplus1}_j$ ($j\\in[1,s]$)      & $\\le\\alpha$       & $\\left(b_i\\to a^{\\text{bin}(i,j)\\oplus1}_j\\right)$ \\\\\n\t\t\t\t\t\t\t\t\t\t\t& $a^{\\text{bin}(i,j)}_j$ ($j\\in[1,s]$)             & $\\le2\\alpha$      & $\\left(b_i\\to b_{\\text{adj}(i,j)}\\to a^{\\text{bin}(i,j)}_j\\right)$ \\\\\n\t\t\t\t\t\t\t\t\t\t\t& $a^\\star_j$ ($j\\in[1,\\ell]$)                      & $\\le\\beta$        & $\\left(b_i\\to a^\\star_j\\right)$ \\\\\n\\midrule\nrouter \t\t\t\t\t\t\t\t\t\t& router                                            & $\\le2\\alpha$      & $\\left(u\\to t\\to v\\right)$ \\\\\n\\bottomrule[1.5pt]\n\\end{tabular}\n\\end{table*}\n\nRegarding the distance between $a_i$ and $b_i$ for $i\\in[1,2^s]$, if there exists $j\\in[1,\\ell]$ such that $x_{i,j}=y_{i,j}=1$, then $w(\\{a_i,a^\\star_j\\})=w(\\{b_i,a^\\star_j\\})=\\alpha$ and $d_{G',w}(a_i,b_i)\\le2\\alpha$ because of the path $(a_i\\to a^\\star_j\\to b_i)$ in $G'$.\nIf there is no $j\\in[1,\\ell]$ such that $x_{i,j}=y_{i,j}=1$, we claim that  $d_{G',w}(a_i,b_i)\\geq\\min\\{\\alpha+\\beta, 3\\alpha\\}$.\nFor any path between $a_i$ and $b_i$, if it contains exactly two edges, it is of the form $(a_i\\to a^\\star_j\\to b_i)$ for some $j\\in[1,\\ell]$ by the construction of $G'$, and it is of length at least $\\alpha+\\beta$ by the assumption.\nIf it contains at least three edges, it is of length at least $3\\alpha$.\n\nIf $F(x,y)=1$, then for any $i\\in[1,2^s]$, there exists $j\\in[1,\\ell]$ such that $x_{i,j}=y_{i,j}=1$.\nHence,\n$$\n\\begin{aligned}\n& d_{G',w}(a_i,b_i)\\le2\\alpha,\\forall i\\in[1,2^s], \\\\\n& D_{G',w}=\\max_{u,v}d_{G',w}(u,v)\\le\\max\\{2\\alpha,\\beta\\}.\n\\end{aligned}\n$$\nTherefore, $D_{G,w}\\le D_{G',w}+n\\le\\max\\{2\\alpha,\\beta\\}+n$ by Lemma~\\ref{lem:contraction}.\n\nIf $F(x,y)=0$, then there exists $i\\in[1,2^s]$ such that $x_{i,j}=0$ or $y_{i,j}=0$ for any $j\\in[1,\\ell]$.\nHence,\n$$\n\\begin{aligned}\n& d_{G',w}(a_i,b_i)=\\min_{\\text{path }P\\text{ from }a_i\\text{ to }b_i}\\text{length}(P)\\ge\\min\\{\\alpha+\\beta,3\\alpha\\}, \\\\\n& D_{G',w}=\\max_{u,v}d_{G',w}(u,v)\\ge d_{G',w}(a_i,b_i)\\ge\\min\\{\\alpha+\\beta,3\\alpha\\}.\n\\end{aligned}\n$$\nTherefore, $D_{G,w}\\ge D_{G',w}\\ge\\min\\{\\alpha+\\beta,3\\alpha\\}$ by Lemma~\\ref{lem:contraction}.\n\\end{proof}\n\nCombining Lemma~\\ref{lem:simulation} and Lemma~\\ref{lem:reduction}, we have a reduction from computing $F$ in the Server model to approximating diameter in the quantum CONGEST model.\nTo prove the communication complexity of $F$ in the Server model, we adopt the following lemma.\n\n\\begin{lemma}[Lemma B.4 in \\cite{ElkinKNP14}, arXiv version]\nFunction ${\\rm VER}:\\{0,1,2,3\\}\\times\\{0,1,2,3\\}\\to\\{0,1\\}$ is defined by ${\\rm VER}(x,y)=1$ if and only if $x+y$ is equivalent to $0$ or $1$ modulo $4$, where $x,y\\in\\{0,1,2,3\\}$.\nLet $f:\\{0,1\\}^k\\to\\{0,1\\}$ be an arbitrary function.\nThen\n$$Q^{sv}_\\varepsilon(f\\circ{\\rm VER}^k)\\ge\\frac12\\text{deg}_{4\\varepsilon}(f)-O(1)$$\nfor any $0<\\varepsilon<1\/4$.\n\\label{lem:lifting}\n\\end{lemma}\n\nA read-once formula, which consists of AND gates, OR gates, and NOT gates, is a formula in which each variable appears exactly once.\nWe will need the following conclusion for approximate degree of read-once formulas.\n\n\\begin{lemma}[Theorem 6 in \\cite{AaronsonBKRT21}]\nFor any read-once formula $f:\\{0,1\\}^k\\to\\{0,1\\}$, $\\text{deg}_{1\/3}(f)=\\Theta\\left(\\sqrt k\\right)$.\n\\label{lem:read_once}\n\\end{lemma}\n\n\\begin{lemma}\nGiven $s,\\ell$ defined in Eq.~\\eqref{eqn:paramters} where $\\ell$ is a multiple of $4$, $F={\\rm AND}_{2^s}\\circ({\\rm OR}_\\ell\\circ{\\rm AND}^\\ell_2)^{2^s}$ with inputs $x,y\\in\\{0,1\\}^{2^s\\cdot\\ell}$, set\n\\[F(x,y)=\\bigwedge_{i\\in[1,2^s]}\\left(\\bigvee_{j\\in[1,\\ell]}\\left(x_{i,j}\\wedge y_{i,j}\\right)\\right).\\]\nIt holds that\n$$Q^{sv}_{1\/12}(F)=\\Omega\\left(\\sqrt{2^s\\cdot\\ell}\\right).$$\n\\label{lem:and_or_and}\n\\end{lemma}\n\\begin{proof}\nThe function $F$ can be rewritten as $F=f\\circ \\text{GDT}^{2^s\\cdot\\ell\/4}$, where\n$f=\\text{AND}_{2^s}\\circ\\text{OR}^{2^s}_{\\ell\/4}$ and $\\text{GDT}=\\text{OR}_4\\circ\\text{AND}^4_2$.\nObviously the function $f$ is a read-once formula.\nIt can be seen that the function VER is actually a {\\it promise version} of the function GDT where inputs $x,y\\in\\{0,1\\}^4$ satisfy\n$$x\\in\\{0011,1001,1100,0110\\},y\\in\\{0001,0010,0100,1000\\}.$$\nThus, the lower bound for $f\\circ\\text{VER}^{2^s\\cdot\\ell\/4}$ clearly implies the lower bound for $f\\circ\\text{GDT}^{2^s\\cdot\\ell\/4}$.\nTherefore,\n$$Q^{sv}_{1\/12}(f\\circ\\text{GDT}^{2^s\\cdot\\ell\/4})\\ge Q^{sv}_{1\/12}(f\\circ\\text{VER}^{2^s\\cdot\\ell\/4})\\ge\\frac12\\text{deg}_{1\/3}(f)-O(1)=\\Omega\\left(\\sqrt{2^s\\cdot\\ell}\\right).$$\nThe second inequality is due to Lemma~\\ref{lem:lifting} and the last inequality is due to Lemma~\\ref{lem:read_once}.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:diameter}]\nLet $\\mathcal A$ be a $T$-round algorithm ($T<2^h\/2$) in the quantum CONGEST model which, for any weighted graph $(G,w)$, computes a $(\\frac32-\\varepsilon)$-approximation of $D_{G,w}$ (constant $\\varepsilon\\in(0,1\/2]$) with probability at least $11\/12$.\nAlice and Bob, who receive $x,y\\in\\{0,1\\}^{2^s\\cdot\\ell}$, respectively, construct the network $G$ as described above with parameters $h,s,\\ell$ given in Eq.~\\eqref{eqn:paramters}.\nThe number of nodes is\n$$n=\\left(2^{h+1}-1\\right)+\\left(2s+\\ell\\right)\\left(2^h+2\\right)+2\\cdot2^s=\\Theta\\left(2^{3h\/2}\\right).$$\nAnd the unweighted diameter is $D_G=\\Theta(h)=\\Theta(\\log n)$.\nLet $w$ be the weight function.\nDue to Lemma~\\ref{lem:simulation}, they can simulate $\\mathcal A$ on $(G,w)$ in the quantum Server model with communication complexity $O(T\\cdot h\\cdot B)$ where $B$ denotes the bandwidth.\nWith probability at least $\\frac{11}{12}$, Alice and Bob output an approximation $\\widetilde D_{G,w}$ satisfying $D_{G,w}\\le\\widetilde D_{G,w}\\le(\\frac32-\\varepsilon)D_{G,w}$.\nWe set $\\alpha=n^2$ and $\\beta=2n^2$.\nBy Lemma~\\ref{lem:reduction},\n$$\n\\begin{aligned}\n\\text{if }F(x,y)=1,\\widetilde D_{G,w} & \\le\\left(\\frac32-\\varepsilon\\right)D_{G,w}\\le\\left(\\frac32-\\varepsilon\\right)\\left(\\max\\{2\\alpha,\\beta\\}+n\\right) \\\\\n& =3n^2-\\left(2\\varepsilon n^2-\\left(\\frac32-\\varepsilon\\right)n\\right); \\\\\n\\text{if }F(x,y)=0,\\widetilde D_{G,w} & \\ge D_{G,w}\\ge\\min\\{\\alpha+\\beta,3\\alpha\\}=3n^2. \\\\\n\\end{aligned}\n$$\nFor large enough $n$, Alice and Bob can distinguish whether $F(x,y)=1$ or not with probability at least $\\frac{11}{12}$ in the Server model, and thus $Q^{sv}_{1\/12}(F)=O(T\\cdot h\\cdot B)$.\nDue to Lemma~\\ref{lem:and_or_and},\n$$T=\\Omega\\left(\\frac{\\sqrt{2^s\\cdot\\ell}}{h\\cdot B}\\right)=\\Omega\\left(\\frac{2^h}{h\\cdot B}\\right)=\\Omega\\left(\\frac{n^{2\/3}}{ \\log^2 n}\\right),$$\nwhere the last equality is by the choice of $h$ and the the bandwidth $B=\\Theta(\\log n)$.\nTherefore, the round complexity of approximating diameter is $\\Omega\\left(\\min\\left\\{2^h\/2,\\frac{n^{2\/3}}{\\log^2n}\\right\\}\\right)=\\Omega\\left(\\frac{n^{2\/3}}{\\log^2n}\\right)$.\n\\end{proof}\n\n\n\n\\subsection{Hardness of Approximating Radius}\n\nWe choose the same set of parameters $h,s,\\ell$ given in Eq.~\\eqref{eqn:paramters}.\nThe argument is very close to the one for diameter.\n\n\\begin{theorem}[Restated]\nFor any constant $\\varepsilon\\in(0,\\frac12]$, any algorithm, with probability at least $\\frac{11}{12}$, computing a $(\\frac32-\\varepsilon)$-approximation of radius in the quantum CONGEST model requires $\\Omega\\left(\\frac{n^{2\/3}}{\\log^2n}\\right)$ rounds, even when the unweighted diameter is $\\Theta(\\log n)$, where $n$ denotes the number of nodes.\n\\label{thm:radius}\n\\end{theorem}\n\nThe weighted graph $(G,w)$ that we construct for showing hardness of approximating radius is almost the same except that we add a node $a_0$ in $V_A$ along with edges $\\{a_0,a_1\\},\\cdots,\\{a_0,a_{2^s}\\}$ of weight $2\\alpha$.\nHere we only show in Figure~\\ref{fig:radius} the graph $G'$ after contracting all edges of weight $1$ (the green edges are the new-added edges).\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.6\\linewidth]{figures\/Radius.pdf}\n\\caption{\nGraph $G'$ (after contraction) for approximating radius.\nThe additional green edges are of weight $2\\alpha$.\nThe eccentricity of any node, except $a_i$ for $i\\in[1,2^s]$, is at least $3\\alpha$; and the eccentricity of $a_i$ is at most $\\max\\{2\\alpha,\\beta\\}$ if there exists $j\\in[1,\\ell]$ such that $x_{i,j}=y_{i,j}=1$, otherwise it is at least $\\min\\{\\alpha+\\beta,3\\alpha\\}$.\nTherefore, the radius is at most $\\max\\{2\\alpha,\\beta\\}$ if there exist $i\\in[1,2^s],j\\in[1,\\ell]$ such that $x_{i,j}=y_{i,j}=1$, otherwise it is at least $\\min\\{\\alpha+\\beta,3\\alpha\\}$.\n}\n\\label{fig:radius}\n\\end{figure}\n\nFor inputs $x,y\\in\\{0,1\\}^{2^s\\cdot\\ell}$ define\n$$F'(x,y)=\\bigvee_{i\\in[1,2^s],j\\in[1,\\ell]}(x_{i,j}\\wedge y_{i,j}).$$\nWe have the following lemma.\n\n\\begin{lemma}\n$R_{G,w}\\le\\max\\{2\\alpha,\\beta\\}+n$ if $F'(x,y)=1$, and $R_{G,w}\\ge\\min\\{\\alpha+\\beta,3\\alpha\\}$ otherwise.\n\\label{lem:reduction_radius}\n\\end{lemma}\n\\begin{proof}\nIt suffices to estimate the radius of $(G',w)$ by Lemma~\\ref{lem:contraction}.\nFor any node $v\\notin\\{a_0,a_1,\\cdots,a_{2^s}\\}$, $d_{G',w}(a_0,v)\\ge3\\alpha$.\nThis is because that any path from $a_0$ to $v$ is of the form $(a_0\\to a_i\\leadsto v)$ for some $i\\in[1,2^s]$, where $w(\\{a_0,a_i\\})=2\\alpha$, and the remaining edges on the path have total weight at least $\\alpha$.\nTherefore, $e_{G',w}(v)\\ge3\\alpha$ for any $v\\notin\\{a_1,\\cdots,a_{2^s}\\}$.\nTo estimate the eccentricity of $a_i$ for $i\\in[1,2^s]$, we have $d(a_i,v)\\le\\max\\{2\\alpha,\\beta\\}$ for any $v\\ne b_i$ as shown on Table~\\ref{tab:distance}, and $d_{G',w}(a_i,b_i)\\le2\\alpha$ if there exists $j\\in[1,\\ell]$ such that $x_{i,j}=y_{i,j}=1$, and $d_{G',w}(a_i,b_i)\\ge\\min\\{\\alpha+\\beta,3\\alpha\\}$ otherwise.\n\nIf $F'(x,y)=1$, then there are $i\\in[1,2^s]$ and $j\\in[1,\\ell]$ such that $x_{i,j}=y_{i,j}=1$, and thus\n$$\n\\begin{aligned}\n& d_{G',w}(a_i,b_i)\\le2\\alpha,\\\\\n& e_{G',w}(a_i)=\\max_v d_{G',w}(a_i,v)\\le\\max\\{2\\alpha,\\beta\\}, \\\\\n& R_{G',w}=\\min_u e_{G',w}(u)\\le e_{G',w}(a_i)\\le\\max\\{2\\alpha,\\beta\\}.\n\\end{aligned}\n$$\nTherefore, $R_{G,w}\\le R_{G',w}+n\\le\\max\\{2\\alpha,\\beta\\}+n$ by Lemma~\\ref{lem:contraction}.\n\nIf $F'(x,y)=0$, then for any $i\\in[1,2^s]$ and $j\\in[1,\\ell]$, $x_{i,j}=0$ or $y_{i,j}=0$, and thus\n$$\n\\begin{aligned}\n& d_{G',w}(a_i,b_i)\\ge\\min\\{\\alpha+\\beta,3\\alpha\\},\\forall i\\in[1,2^s], \\\\\n& e_{G',w}(a_i)=\\max_v d_{G',w}(a_i,v)\\ge d_{G',w}(a_i,b_i) \\\\\n& \\qquad\\ge\\min\\{\\alpha+\\beta,3\\alpha\\},\\forall i\\in[1,2^s], \\\\\n& R_{G',w}=\\min_u e_{G',w}(u)\\ge\\min\\{\\alpha+\\beta,3\\alpha\\}.\n\\end{aligned}\n$$\nTherefore, $R_{G,w}\\ge R_{G',w}\\ge\\min\\{\\alpha+\\beta,3\\alpha\\}$ by Lemma~\\ref{lem:contraction}.\n\\end{proof}\n\nSimilar to Lemma~\\ref{lem:and_or_and}, one can prove a lower bound on communication complexity of $F'$ in the quantum Server model.\n\n\\begin{lemma}\nGiven $s,\\ell$ defined in Eq.~\\eqref{eqn:paramters} where $2^s\\cdot\\ell$ is a multiple of $4$,\n$F'={\\rm OR}_{2^s\\cdot\\ell}\\circ{\\rm AND}^{2^s\\cdot\\ell}_2$ with inputs $x,y\\in\\{0,1\\}^{2^s\\cdot\\ell}$,\nset\n$$F'(x,y)=\\bigvee_{i\\in[1,2^s],j\\in[1,\\ell]}(x_{i,j}\\wedge y_{i,j}).$$\nIt holds that\n$$Q^{sv}_{1\/12}(F')=\\Omega\\left(\\sqrt{2^s\\cdot\\ell}\\right).$$\n\\label{lem:or_and}\n\\end{lemma}\n\\begin{proof}\nThe function $F'$ can be rewritten as $F'=f'\\circ \\text{GDT}^{2^s\\cdot\\ell\/4}$,\nwhere $f'=\\text{OR}_{2^s\\cdot\\ell\/4}$.\nNote that $f'$ is still a read-once formula.\nThus the rest of proof is the same as the one in Lemma~\\ref{lem:and_or_and}.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:radius}]\nLet $\\mathcal A$ be a $T$-round algorithm ($T<2^h\/2$) in the quantum CONGEST model which, for any weighted graph $(G,w)$, computes a $(\\frac32-\\varepsilon)$-approximation of $R_{G,w}$ (constant $\\varepsilon\\in(0,\\frac12]$) with probability at least $\\frac{11}{12}$.\nAlice and Bob, who receive $x,y\\in\\{0,1\\}^{2^s\\cdot\\ell}$ as input, construct the weighted graph $(G,w)$ described above with the number of node $n=\\Theta(2^{3h\/2})$. The unweighted diameter $D_G=\\Theta(\\log n)$.\nDue to Lemma~\\ref{lem:simulation}, Alice and Bob can simulate $\\mathcal A$ on $(G,w)$ in the quantum Server model with communication complexity $O(T\\cdot h\\cdot B)$.\nThen with probability at least $\\frac{11}{12}$, Alice and Bob compute $\\widetilde R_{G,w}$ satisfying $R_{G,w}\\le\\widetilde R_{G,w}\\le(\\frac32-\\varepsilon)R_{G,w}$.\nWe set $\\alpha=n^2$ and $\\beta=2n^2$.\nDue to Lemma~\\ref{lem:reduction_radius},\n$$\n\\begin{aligned}\n& \\text{if }F'(x,y)=1,\\widetilde R_{G,w}\\le3n^2-\\left(2\\varepsilon n^2-\\left(\\frac32-\\varepsilon\\right)n\\right); \\\\\n& \\text{if }F'(x,y)=0,\\widetilde R_{G,w}\\ge3n^2. \\\\\n\\end{aligned}\n$$\nFor large enough $n$, Alice and Bob can compute $F'$ with probability at least $\\frac{11}{12}$ in the Server model, and thus $Q^{sv}_{1\/12}(F')=O(T\\cdot h\\cdot B)$.\nDue to Lemma~\\ref{lem:or_and}, $T=\\Omega\\left(\\frac{n^{2\/3}}{\\log^2n}\\right)$.\nTherefore, the round complexity of approximating radius is $\\Omega\\left(\\min\\left\\{2^h\/2,\\frac{n^{2\/3}}{\\log^2n}\\right\\}\\right)=\\Omega\\left(\\frac{n^{2\/3}}{\\log^2n}\\right)$.\n\\end{proof}\n\\section{Introduction}\n\nQuantum distributed computing has received great attention in the past decade~\\cite{Ben-OrH05,TaniKM12,ElkinKNP14,GallM18,GallNR19,IzumiG19,IzumiGM20,Censor-HillelFG22,MagniezN20,GavoilleKM09}.\nA large body of work has been devoted to investigating the quantum advantages in distributed computing.\nIn this paper, we are concerned with the CONGEST networks, which are one of the most fundamental models in distributed computing.\nIn a classical CONGEST network, the nodes synchronously exchange classical messages, and each channel has $O(\\log n)$-bit bandwidth, where $n$ is the number of nodes in the network.\nQuantum CONGEST networks were first introduced by  Elkin, Klauck, Nanongkai,\nand Pandurangan~\\cite{ElkinKNP14}, where the only difference is that the nodes exchange quantum messages and the bandwidth of each channel is $O(\\log n)$ qubits. The round complexity of diameter and radius  of unweighted graphs in classical CONGEST networks has been extensively studied~\\cite{AbboudCK16,AnconaCDEW20,HolzerW12,PelegRT12,FrischknechtHW12,HolzerPRW14}.\nLe Gall and Magniez~\\cite{GallM18} proved that quantum communication may save the round complexity in CONGEST networks if the graph has a low diameter.\n\nIn this paper, we further investigate the round complexity of computing the diameters and radius of weighted graphs in quantum CONGEST networks. We prove that quantum communication may also save the round complexity for both problems.\n\n\n\n\\subsection{Our Results}\n\\label{sec:results}\n\nThe following is one of our main results which asserts that quantum communication may save the round complexity for computing the weighted diameter and radius of a graph given that the graph has a low unweighted diameter.\n\n\\begin{theorem}\nThere exists a $\\widetilde O\\left(\\min\\left\\{n^{9\/10}D^{3\/10},n\\right\\}\\right)$-round distributed algorithm computing a $(1+o(1))$-approximation of the weighted diameter\/radius with probability at least $1-1\/\\text{poly}(n)$, in the quantum CONGEST model, where $D$ denotes the unweighted diameter.\n\\label{thm:upper_bound}\n\\end{theorem}\n\nHolzer and Pinsker in~\\cite{HolzerP15} and Abboud, Censor-Hillel and Khoury in~\\cite{AbboudCK16} proved that $(2-o(1))$-approximating the diameter and $(3\/2-\\varepsilon)$-approximating the (even unweighted) radius in the classical CONGEST network require $\\widetilde\\Omega(n)$ rounds, even when $D$ is constant.\nTherefore, Theorem~\\ref{thm:upper_bound} exhibits the advantages of quantum communication over classical communication in approximating the weighted diameter\/radius when $D=o(n^{1\/3})$.\n\nWe prove Theorem~\\ref{thm:upper_bound} by applying the framework of distributed quantum optimization introduced by Le Gall and Magniez in~\\cite{GallM18}.\nNote that the diameter and radius are the maximum and minimum of eccentricities respectively.\nIt will not give a sublinear-time algorithm if we simply apply a quantum search algorithm, because evaluating the eccentricity of one node takes $\\widetilde\\Theta\\left(\\sqrt n\\right)$ rounds (the lower bound is due to \\cite{ElkinKNP14}), and the searching process should require another $\\widetilde\\Theta\\left(\\sqrt n\\right)$ times of evaluation (the number of nodes with maximum\/minimum eccentricity maybe $O(1)$).\nThus the number of rounds in total will be $\\widetilde{\\Theta}(n)$.\\footnote{\n$\\widetilde O(\\cdot)$, $\\widetilde\\Omega(\\cdot)$, and $\\widetilde\\Theta(\\cdot)$ hide polylogarithmic factors.}\nOur algorithm is inspired by Nanongkai's algorithm~\\cite{Nanongkai14STOC} for approximating the weighted shortest paths in a classical network.\nThe algorithm constructs several small vertex sets and searches the node achieving the maximum\/minimum eccentricity within those sets, which turns out to be a good approximation of diameter\/radius.\nOur algorithm quantizes Nanongkai's algorithm using the standard technique \\cite{Bennett89} and further combines with the framework of distributed quantum optimization in~\\cite{GallM18}.\n\nWe also prove lower bounds for approximating weighted diameter and radius.\n\n\\begin{theorem}\nAny algorithm computing a $(3\/2-o(1))$-approximation of the weighted diameter\/radius requires $\\widetilde\\Omega(n^{2\/3})$ rounds, in the quantum CONGEST model, even when $D=\\Theta(\\log n)$.\n\\label{thm:lower_bound_raw}\n\\end{theorem}\n\nThe hardness of both problems is proved via the communication complexity of quantum Server models.\nThe Server model is a variant of two-party communication complexity models introduced in \\cite{ElkinKNP14}.\nCombining with the graph gadget in \\cite{AbboudCK16}, we get a reduction from the communication complexity of certain read-once functions to the round complexity of approximating the weighted diameter and radius.\nWe further apply a lifting theorem of quantum communication complexity~\\cite{ElkinKNP14} to obtain the desired lower bounds.\n\nCompared with Le Gall and Magniez's algorithm~~\\cite{GallM18} for unweighted diameter\/radius with $\\widetilde O\\left(\\sqrt{nD}\\right)$ rounds,  Theorem~\\ref{thm:lower_bound_raw} says that computing weighted diameter\/radius is strictly harder than unweighted diameter\/radius, when $D$ is small.\nWhile in the classical setting, computing weighted and unweighted diameter\/radius have the same round complexity $\\Theta(n)$~\\cite{AbboudCK16,BernsteinN19}\\footnote{The lower bound is proved in~\\cite{AbboudCK16}. The upper bound is followed from the $\\widetilde O(n)$-round algorithm for the exact weighted All-Pairs Shortest Paths (APSP) in~\\cite{BernsteinN19}.}.\n\n\n\n\\subsection{Related Works}\n\nA series of works started with the distance computation in the classical CONGEST network.\nEarlier, Frischknecht, Holzer, and Wattenhofer~\\cite{FrischknechtHW12} showed that computing the diameter of an unweighted graph with constant diameter requires $\\widetilde\\Omega(n)$ rounds, which is tight up to logarithmic factors since even computing All-Pairs Shortest Paths (ASAP) on an unweighted graph can be resolved in $O(n)$ rounds \\cite{HolzerW12,PelegRT12}.\nAbboud, Censor-Hillelet, and Khoury~\\cite{AbboudCK16} later gave the same lower bound of $\\widetilde\\Omega(n)$ for $(3\/2-\\varepsilon)$-approximating the diameter\/radius in sparse networks.\nBernstein and Nanongkai~\\cite{BernsteinN19} provided a $\\widetilde O(n)$-round algorithm computing the exact APSP on any weighted graph.\nAs a result, computing unweighted diameter\/radius and weighted diameter\/radius (exactly or with a small approximation ratio) have an almost tight complexity of $\\widetilde\\Theta(n)$ in the classical CONGEST network.\nIf a larger approximation ratio is allowed, there are $\\widetilde O(\\sqrt n+D)$-round algorithms for $3\/2$-approximating the diameter\/radius on any unweighted graph \\cite{HolzerPRW14,AnconaCDEW20}.\nBesides, Chechik and Mukhtar~\\cite{ChechikM20} showed a $\\widetilde O(\\sqrt nD^{1\/4}+D)$-round algorithm computing Single-Source Shortest Paths (SSSP) exactly on any weighted graph, which also gives a $2$-approximation of the diameter\/radius.\n\nAs for the quantum setting, while quantum computation offers advantages over classical computation in various settings such as query complexity and two-party communication complexity, the power of quantum computation in distributed computing has not been fully explored.\nIn the quantum CONGEST network, Elkin et al.~\\cite{ElkinKNP14} gave negative results for several problems such as minimum spanning tree, minimum cut, and SSSP, i.e., quantum communication does not speed up distributed algorithms for these problems.\nLe~Gall and Magniez~\\cite{GallM18} presented a $\\widetilde O\\left(\\sqrt{nD}\\right)$-round algorithm computing the diameter\/radius on any unweighted graph, along with a $\\widetilde O\\left(\\sqrt[3]{nD}+D\\right)$-round algorithm $3\/2$-approximating the diameter.\nThey also proved a $\\widetilde\\Omega(\\sqrt n+D)$-lower bound for computing the unweighted diameter, which was later improved to $\\widetilde\\Omega\\left(\\sqrt[3]{nD^2}+\\sqrt n\\right)$ by Magniez and Nayak~\\cite{MagniezN20}.\nThe above results are listed on Table~\\ref{tab:complexity}.\n\n\\begin{table}[t]\n\\centering\n\\scriptsize\n\\renewcommand\\arraystretch{1.5}\n\\caption{Complexity of computing diameter and radius in the CONGEST model.}\n\\label{tab:complexity}\n\\begin{tabular}{cccccccc}\n\\toprule[1.5pt]\n\\multirow{2}{*}{Problem} & \\multirow{2}{*}{Variant} & \\multirow{2}{*}{Approx.} & \\multicolumn{2}{c}{Upper bound $\\widetilde O(\\cdot)$} & ~ & \\multicolumn{2}{c}{Lower bound $\\widetilde\\Omega(\\cdot)$} \\\\\n\\cline{4-5} \\cline{7-8}\n~ & ~ & ~ & Classical & Quantum & ~ & Classical & Quantum \\\\\n\\midrule[1.5pt]\n\\multirow{7}{*}{diameter} & ~ & exact & $n$~\\cite{HolzerW12,PelegRT12} & $\\sqrt{nD}$~\\cite{GallM18} & ~ & $n$~\\cite{FrischknechtHW12} & $\\sqrt[3]{nD^2}+\\sqrt n$~\\cite{MagniezN20} \\\\\n~ & ~ & $3\/2-\\varepsilon$ & $n$ & $\\sqrt{nD}$ & ~ & $n$~\\cite{AbboudCK16} & $\\sqrt n+D$~\\cite{GallM18} \\\\\n~ & ~ & $3\/2$ & $\\sqrt n+D$~\\cite{HolzerPRW14,AnconaCDEW20} & $\\sqrt[3]{nD}+D$~\\cite{GallM18} & ~ & open & open \\\\\n~ & weighted & exact & $n$~\\cite{BernsteinN19} & $n$ & ~ & $n$ & $n^{2\/3}$ \\\\\n~ & weighted & $(1,3\/2)$ & $n$ & \\textcolor{red}{$\\min\\left\\{n^{9\/10}D^{3\/10},n\\right\\}$ (This work)} & ~ & $n$ & \\textcolor{red}{$n^{2\/3}$ (This work)} \\\\\n~ & weighted & $2-\\varepsilon$ & $n$ & $\\min\\left\\{n^{9\/10}D^{3\/10},n\\right\\}$ & ~ & $n$~\\cite{HolzerP15} & $\\sqrt n+D$ \\\\\n~ & weighted & $2$ & $\\sqrt nD^{1\/4}+D$~\\cite{ChechikM20} & $\\sqrt nD^{1\/4}+D$ & ~ & open & open \\\\\n\\midrule\n\\multirow{6}{*}{radius} & ~ & exact & $n$~\\cite{HolzerW12,PelegRT12} & $\\sqrt{nD}$ & ~ & $n$ & $\\sqrt[3]{nD^2}+\\sqrt n$ \\\\\n~ & ~ & $3\/2-\\varepsilon$ & $n$ & $\\sqrt{nD}$ & ~ & $n$~\\cite{AbboudCK16} & $\\sqrt n+D$ \\\\\n~ & ~ & $3\/2$ & $\\sqrt n+D$~\\cite{AnconaCDEW20} & $\\sqrt n+D$ & ~ & open & open \\\\\n~ & weighted & exact & $n$~\\cite{BernsteinN19} & $n$ & ~ & $n$ & $n^{2\/3}$ \\\\\n~ & weighted & $(1,3\/2)$ & $n$ & \\textcolor{red}{$\\min\\left\\{n^{9\/10}D^{3\/10},n\\right\\}$ (This work)} & ~ & $n$ & \\textcolor{red}{$n^{2\/3}$ (This work)} \\\\\n~ & weighted & $2$ & $\\sqrt nD^{1\/4}+D$~\\cite{ChechikM20} & $\\sqrt nD^{1\/4}+D$ & ~ & open & open \\\\\n\\bottomrule[1.5pt]\n\\end{tabular}\n\\end{table} \n\\section{Preliminaries}\n\n\\subsection{Graph Notations}\n\\label{sec:graph_notations}\n\nGiven a weighted graph $(G,w)$ where $G=(V,E)$ and $w:E\\to\\mathbb N^+$.\nThe length of a path is defined to be the sum of weights of edges on it, and the distance between nodes $u$ and $v$, denoted by $d_{G,w}(u,v)$, is the least length over all paths between them.\nThe eccentricity of a node $u$ is denoted by $e_{G,w}(u)=\\max_{v\\in V}d_{G,w}(u,v)$.\nThe radius of weighted graph $(G,w)$, denoted by $R_{G,w}$, is the minimum eccentricity over all nodes, i.e., $R_{G,w}=\\min_{u\\in V}e_{G,w}(u)$, while the diameter of $(G,w)$, denoted by $D_{G,w}$, is the maximum eccentricity of nodes, or equally, the maximum distance between any two nodes, i.e., $D_{G,w}=\\max_{u\\in V}e_{G,w}(u)=\\max_{u,v\\in V}d_{G,w}(u,v)$.\nThe unweighted diameter of graph $G$ is denoted by $D_G=D_{G,w^\\star}$ where $w^\\star(e)=1$ for all $e\\in E$, which is an essential parameter when $G$ represents the underlying graph of a distributed network.\n\n\\subsection{CONGEST Model}\n\nIn the classical CONGEST model, the communication network is a graph $G=(V,E)$ with $n$ nodes, and every node is assigned with a unique identifier.\nEach node represents a processor with unlimited computational power, i.e., the consumption of any local computation in a single processor is ignored.\nEach edge connecting two nodes represents a communication channel with $B=O(\\log n)$ bits of bandwidth.\nIn this article, we further consider the weighted graph $(G,w)$ as underlying network, where the weight of each edge is initially known to both of its endpoints.\n\nFor quantum version of the CONGEST model defined in \\cite{ElkinKNP14}, adjacent nodes are allowed to exchange \\textit{qubits} (quantum bits), i.e., the classical channels are now quantum channels with the same bandwidth $B=O(\\log n)$.\nEach node can locally do some quantum computation, and distinct nodes may own qubits with entanglement.\nIn this paper, we assume that initially all nodes do not share any entanglement, but the nodes can, for example, locally create a pair of entangled qubits, and send one to others.\n\nFor both classical and quantum CONGEST models, the algorithm is implemented round by round in a synchronous manner.\nIn each round, each node sends\/receives a message of $O(\\log n)$ (qu)bits to\/from each neighbor, and then does local computation according to local knowledge.\nThe algorithm halts when all nodes halt, and at the end of the algorithm, each node has its own output. We say an algorithm computes the diameter\/radius if all nodes output the correct answer.\nThe round complexity of an algorithm in this model is defined to be the number of communication rounds needed.\nAnd the round complexity of a distributed problem is the least round complexity of any algorithm solving it.\nOur focus here is the distance problems, mainly the computation of diameter and radius mentioned in Section~\\ref{sec:graph_notations}.\n\n\\subsection{Server Model}\n\nThe Sever model  is a variant of the two-party communication model, which was introduced by Elkin et al.~\\cite{ElkinKNP14} to prove lower bounds in the CONGEST model.\nThere are three players in the Server model: Alice, Bob, and the server.\nAlice and Bob receive the inputs $x$ and $y$ respectively, and want to compute $F(x,y)$ for some function $F$.\nThe server receives no input.\nAlice and Bob can exchange messages with the server.\nThe catch here is that the server can send messages for free.\nThus, the communication complexity counts only messages sent by Alice and Bob.\nNote that Alice and Bob can talk to each other by considering the server as a communication channel, so any protocol in the traditional two-party communication model can be implemented in the Server model with the same complexity.\n\nFor a two-argument function $F$ and $0\\le\\varepsilon<1$, we let $Q^{sv}_\\varepsilon(F)$ denote the communication complexity (in the quantum setting) of computing $F$ where for any inputs $x,y$, the algorithm must output $F(x,y)$ with probability at least $1-\\varepsilon$.\nFor Boolean function $f:\\{0,1\\}^n\\rightarrow\\mathbb{R}$ and $0\\le\\varepsilon<1$, the $\\varepsilon$-approximate degree of $f$, denoted by $\\text{deg}_\\varepsilon(f)$, is the smallest degree of any polynomial $p$ that $\\varepsilon$-represents $f$, i.e., $|p(x)-f(x)|\\le\\varepsilon$ for any input $x\\in\\{0,1\\}^n$.\n\\section{Algorithm}\n\nWe first introduce the framework of distributed quantum optimization in \\cite{GallM18}.\nGiven function  $f:X\\to\\mathbb Z$, where $X$ is a finite set, let $G=(V,E)$ be a network with a pre-defined node $\\text{leader}\\in V$.\nWe write $\\ket\\phi_v$ to denote a state in the memory space of node $v$.\nA specific register $\\ket\\cdot_I$ called \\textit{internal} and the control of the algorithm are centralized by the node leader.\nAssume that the following three quantum procedures are given as black boxes.\n\\begin{itemize}\n\\item {\\bf Initialization:} Prepare an initial state $\\ket0_I\\ket{\\text{init}}$ with some pre-computed information $\\ket{\\text{init}}$.\n\\item {\\bf Setup:} Produce a superposition from the initial state:\n$$\\ket0_I\\ket{\\text{init}}\\mapsto\\sum_{x\\in X}\\alpha_x\\ket x_I\\ket{\\text{data}(x)}\\ket{\\text{init}},$$\nwhere the $\\alpha_x$'s are arbitrary amplitudes and $\\text{data}(x)$ are information depending on $x$.\n\\item {\\bf Evaluation:} Perform the transformation\n$$\\ket{x,0}_I\\ket{\\text{data}(x)}\\ket{\\text{init}}\\mapsto\\ket{x,f(x)}_I\\ket{\\text{data}(x)}\\ket{\\text{init}}.$$\n\\end{itemize}\n\nThe following lemma provides an algorithm to search $x\\in X$ with high value $f(x)$ given the three procedures above.\n\n\\begin{lemma}[Theorem 2.4 in \\cite{GallM18}\\footnote{Although Le~Gall and Magniez write a slightly weaker statement, the lemma we claim here can be proven by the same argument in \\cite{GallM18}.}]\nAssume that {\\rm Initialization} can be implemented within $T_0$ rounds in the quantum CONGEST model, and that unitary operators {\\rm Setup} and {\\rm Evaluation} and their inverses can be implemented within $T$ rounds.\nLet $\\rho>0$ be such that $\\sum_{x\\in X:f(x)\\ge M}|\\alpha_x|^2\\ge\\rho$ where $M$ is unknown to all nodes.\nThen, for any $\\delta>0$, the node {\\rm leader} can find, with probability at least $1-\\delta$, some element $x$ such that $f(x)\\ge M$, in $T_0+O\\left(\\sqrt{\\log(1\/\\delta)\/\\rho}\\right)\\times T$ rounds.\n\\label{lem:optimization}\n\\end{lemma}\n\nThe three procedures will be described as deterministic or randomized procedures that combine the subroutines provided by Nanongkai~\\cite{Nanongkai14STOC} (also presented in Appendix~\\ref{sec:toolkits}).\nThey can be quantized using the standard technique \\cite{Bennett89}, with potentially additional garbage whose size is of the same order as the initial memory space.\n\nGiven a weighted graph $(G,w)$ where $G=(V,E)$ is a network and $w:E\\to\\mathbb N^+$, we show a quantum algorithm approximating $D_{G,w}$ and $R_{G,w}$ by proving Theorem~\\ref{thm:upper_bound}.\nWe only show the algorithm approximating the diameter.\nThe proof for radius is basically the same except that it finds the minimum (approximate) eccentricity instead of the maximum one.\n\nWe choose the parameters throughout this section.\n\\begin{equation}\n\\label{eqn:alg_parameters}\n\\varepsilon=1\/\\log n,r=n^{2\/5}D_G^{-1\/5},\\ell=n\\log n\/r,k=\\sqrt{D_G}.\n\\end{equation}\nAs mentioned in Section~\\ref{sec:results}, finding a node with maximum eccentricity among all nodes by directly applying a quantum search algorithm can hardly be done in $o(n)$ rounds.\nWe instead try to find a vertex set containing a node with maximum approximate eccentricity among $n$ vertex sets $S_1,\\cdots,S_n$, and then search such a node in this vertex set.\nEach set $S_i$ for $i\\in[1,n]$ is sampled by having each node $v\\in V$ join it independely with probability $r\/n$.\nFor such a random set and a node $s$ in it, Nanongkai showed in \\cite{Nanongkai14STOC} an efficient classical procedure to approximate its eccentricity (actually every node $v\\in V$ can know an approximation of the distance from $s$ to $v$).\n\n\n\n\\subsection{Computation of Approximate Eccentricity}\n\\label{sec:approx_ecc}\n\nFor convenience, we need to introduce several graph notations.\nGiven a weighted graph $(G,w)$, the hop distance between nodes $u$ and $v$, denoted by $h_{G,w}(u,v)$, is the minimum number of edges over all shortest paths between them.\nThe hop diameter of the weighted graph, denoted by $H_{G,w}$, is the maximum hop distance between any two nodes.\nFor $\\ell>0$, the $\\ell$-hop distance between $u$ and $v$, denoted by $d^\\ell_{G,w}(u,v)$, is the least length over all paths between them containing at most $\\ell$ edges.\nNote that $d^\\ell_{G,w}(u,v)=d_{G,w}(u,v)$ when $h_{G,w}(u,v)\\le\\ell$.\n\nIn general, Nanongkai~\\cite{Nanongkai14STOC} would approximate the bounded-hop distance, and sample a random set of key nodes as skeleton.\nThen it could approximate the distance from any key node $s$ to any node $v$ since, with high probability, any shortest path from $s$ to $v$ can be partitioned into bounded-hop shortest paths between key nodes, along with a tail path from some key node to $v$, as long as the number of key nodes is sufficiently large.\n\nHere we only list the necessary definitions of approximate bounded-hop distance, approximate distance, and approximate eccentricity.\nWe claim that these are good approximations.\nThe algorithms evaluating these quantities are presented in Appendix~\\ref{sec:toolkits}, and the detailed proof should be found in \\cite[arXiv version]{Nanongkai14STOC}.\nNote that we are given a weighted graph $(G,w)$ where $G=(V,E)$ and $w:E\\to\\mathbb N^+$.\n\n\\begin{lemma}[Theorem 3.3 in \\cite{Nanongkai14STOC}]\nGiven an integer $\\ell>0$.\nFor integer $i\\ge0$, define $w_i:E\\to\\mathbb N^+$ where $w_i(e)=\\left\\lceil\\frac{2\\ell w(e)}{\\varepsilon\\cdot2^i}\\right\\rceil$ for $e\\in E$.\nFor any $u,v\\in V$, the approximate bounded-hop distance is defined as\n$$\\widetilde d^\\ell_{G,w}(u,v)=\\min_i\\left\\{d_{G,w_i}(u,v)\\cdot\\frac{\\varepsilon\\cdot2^i}{2\\ell}:d_{G,w_i}(u,v)\\le\\left(1+\\frac2\\varepsilon\\right)\\ell\\right\\}.$$\nThen $d_{G,w}(u,v)\\le\\widetilde d^\\ell_{G,w}(u,v)\\le(1+\\varepsilon)d^\\ell_{G,w}(u,v)$.\n\\label{lem:bounded_hop_distance}\n\\end{lemma}\n\n\\begin{lemma}[Theorem 4.2 in \\cite{Nanongkai14STOC}]\nGiven a vertex set $S\\subseteq V$.\nLet the weighted complete graph $\\left(G'_S,w'_S\\right)$ be such that\n$$\n\\begin{aligned}\n& G'_S=\\left(S,\\tbinom{S}{2}\\right),w'_S:\\tbinom{S}{2}\\to\\mathbb N^+, \\\\\n& w'_S(\\{u,v\\})=\\widetilde d^\\ell_{G,w}(u,v),\\forall\\{u,v\\}\\in \\tbinom{S}{2}.\n\\end{aligned}\n$$\nFor node $v\\in S$, let $N^k_S(v)$ be the set of the $k$ nodes with the least distance from $v$ on $(G'_S,w'_S)$.\nAnd let the weighted complete graph $\\left(G''_S,w''_S\\right)$ be such that\n$$\n\\begin{aligned}\n& G''_S=\\left(S,\\tbinom{S}{2}\\right),w''_S:\\tbinom{S}{2}\\to\\mathbb N^+, \\\\\n& w''_S(\\{u,v\\})=\n\\begin{cases}\nd_{G'_S,w'_S}(u,v),\t& u\\in N^k_S(v)\\text{ or }v\\in N^k_S(u) \\\\\nw'_S(\\{u,v\\}), \t& \\text{otherwise}\n\\end{cases},\\forall\\{u,v\\}\\in \\tbinom{S}{2}.\n\\end{aligned}\n$$\nFor any $s\\in S$ and $v\\in V$, the approximate distance is defined as\n$$\\widetilde d_{G,w,S}(s,v)=\\min_{u\\in S}\\left\\{\\widetilde d^{4|S|\/k}_{G''_S,w''_S}(s,u)+\\widetilde d^\\ell_{G,w}(u,v)\\right\\}.$$\nIf $\\ell=n\\log n\/r$ and $S$ is sampled by having each node $v\\in V$ join it independetly with probability $r\/n$, then $d_{G,w}(s,v)\\le\\widetilde d_{G,w,S}(s,v)\\le(1+\\varepsilon)^2d_{G,w}(s,v)$ for all $s\\in S$ and $v\\in V$, with probability at least $1-2^{-cn}$, for some constant $c>0$ and sufficiently large $n$.\n\\label{lem:approx_distance}\n\\end{lemma}\n\n\\noindent\n{\\bf Remark.} We briefly explain why $\\widetilde d_{G,w,S}(\\cdot)$ is a good approximation.\nBy the choice of $\\ell$ and $S$, Lemma 4.3 in \\cite{Nanongkai14STOC} says that, with high probability, any $s\\in S,v\\in V$ and shortest path $\\left(s\\leadsto v\\right)$ on $(G,w)$ is of the form $\\left(s=s_1\\leadsto\\cdots\\leadsto s_m=u\\leadsto v\\right)$ such that $s_i\\in S$ for $i\\in[1,m]$, $h_{G,w}(s_{i-1},s_i)\\le\\ell$ for $i\\in[2,m]$, and $h_{G,w}(u,v)\\le\\ell$.\nApparently $\\widetilde d_{G,w,S}\\ge d_{G,w}(s,v)$.\nOn the other side,\n\\begin{equation*}\n\\begin{split}\n\\widetilde d_{G,w,S}(s,v) & =\\widetilde d^{4|S|\/k}_{G''_S,w''_S}(s,u)+\\widetilde d^\\ell_{G,w}(u,v) \\\\\n& \\le(1+\\varepsilon)d^{4|S|\/k}_{G''_S,w''_S}(s,u)+\\widetilde d^\\ell_{G,w}(u,v) \\\\\n& =(1+\\varepsilon)d_{G''_S,w''_S}(s,u)+\\widetilde d^\\ell_{G,w}(u,v) \\\\\n& \\le(1+\\varepsilon)\\sum_{i=2}^mw'_S(\\{s_{i-1},s_i\\})+\\widetilde d^\\ell_{G,w}(u,v) \\\\\n& \\le(1+\\varepsilon)\\sum_{i=2}^m\\widetilde d^\\ell_{G,w}(s_{i-1},s_i)+\\widetilde d^\\ell_{G,w}(u,v) \\\\\n& \\le(1+\\varepsilon)^2\\left(\\sum_{i=2}^md^\\ell_{G,w}(s_{i-1},s_i)+d^\\ell_{G,w}(u,v)\\right) \\\\\n& =(1+\\varepsilon)^2\\left(\\sum_{i=2}^md_{G,w}(s_{i-1},s_i)+d_{G,w}(u,v)\\right) \\\\\n& =(1+\\varepsilon)^2d_{G,w}(s,v).\n\\end{split}\n\\end{equation*}\nThe second and sixth lines are due to Lemma~\\ref{lem:bounded_hop_distance}.\nThe third line is due to Theorem 3.10 in \\cite{Nanongkai14STOC}, which says that $H_{G''_S,w''_S}<4|S|\/k$ since $(G''_S,w''_S)$ is the {\\it $k$-shortcut graph} of $(G'_S,w'_S)$.\n\n\\bigskip\n\nFor $i\\in[1,n]$, we rewrite $G''_{S_i},w''_{S_i},\\widetilde d_{G,w,S_i}(\\cdot)$ as $G''_i,w''_i,\\widetilde d_{G,w,i}(\\cdot)$ for short.\nFor any $s\\in S_i$, the approximate eccentricity is defined as $\\widetilde e_{G,w,i}(s)=\\max_{v\\in V}\\widetilde d_{G,w,i}(s,v)$.\nDefine two good events:\n\\begin{itemize}\n\\item {\\bf Good-Scale:} For all $i\\in[1,n]$, $|S_i|=\\Theta(r)$.\nBesides, let $v^\\star\\in V$ be a node with maximum eccentricity, i.e., $e_{G,w}(v^\\star)=D_{G,w}$, then $v^\\star$ joins $\\beta=\\Theta(r)$ sets $S_{i_1},\\cdots,S_{i_\\beta}$.\n\\item {\\bf Good-Approximation:} For all $i\\in[1,n]$ and $s\\in S_i,v\\in V$, $d_{G,w}(s,v)\\le\\widetilde d_{G,w,i}(s,v)\\le(1+\\varepsilon)^2d_{G,w}(s,v)$, thus $e_{G,w}(s)\\le\\widetilde e_{G,w,i}(s)\\le(1+\\varepsilon)^2e_{G,w}(s)$.\n\\end{itemize}\nBy Chernoff inequality and a union bound, the event Good-Scale occurs with probability at least $1-1\/\\text{poly}(n)$.\nBy Lemma~\\ref{lem:approx_distance}\nand a union bound, the event Good-Approximation occurs with probability at least $1-1\/\\text{poly}(n)$.\nTherefore, we can assume that the two events all happen in the following context.\n\n\n\n\\section{Quantization}\n\nFor each $i\\in[1,n]$, we define $f_i:S_i\\to\\mathbb Z$ where $f_i(s)=\\widetilde e_{G,w,i}(s)$ for $s\\in S_i$, and  $f:[1,n]\\to\\mathbb Z$ where $f(i)=\\max_{s\\in S_i}f_i(s)$ for $i\\in[1,n]$.\n\n\\begin{lemma}\nThe number of $i\\in[1,n]$ satisfying $f(i)\\ge D_{G,w}$ is $\\Theta(r)$.\nMoreover, $f(i)\\le(1+\\varepsilon)^2D_{G,w}$  for all $i\\in[1,n]$.\n\\label{lem:approximation}\n\\end{lemma}\n\n\\begin{proof}\n$$\n\\begin{aligned}\n& f(i_j)=\\max_{s\\in S_{i_j}}\\widetilde e_{G,w,i_j}(s)\\ge\\max_{s\\in S_{i_j}}e_{G,w}(s)\\ge e_{G,w}(v^\\star)=D_{G,w}, & \\forall j\\in[1,\\beta]; \\\\\n& f(i)=\\max_{s\\in S_i}\\widetilde e_{G,w,i}(s)\\le\\max_{s\\in S_i}(1+\\varepsilon)^2e_{G,w}(s)\\le(1+\\varepsilon)^2D_{G,w}, & \\forall i\\in[1,n].\n\\end{aligned}\n$$\n\\end{proof}\n\n\\begin{lemma}\nGiven $i\\in[1,n]$, there exists a quantum procedure performing the transformation\n$$\\bigotimes_{v\\in V}\\ket i_v\\ket0_{{\\rm leader}}\\mapsto\\bigotimes_{v\\in V}\\ket i_v\\ket{f(i)}_{{\\rm leader}}$$\nin the quantum CONGEST model, and taking $\\widetilde O\\left(D_G+\\frac n{\\varepsilon\\cdot r}+rk+\\sqrt r\\left(\\frac r{\\varepsilon\\cdot k}\\cdot D_G+r\\right)\\right)$ rounds, with probability at least $1-1\/{\\rm poly}(n)$.\n\\label{lem:evaluation}\n\\end{lemma}\n\n\\begin{proof}\nWe give the quantum procedure maximizing $f_i$ (thus evaluating $f(i)$) by following the framework of distributed quantum optimization:\n\\begin{itemize}\n\\item {\\bf $\\text{Initialization}_i$:} Perform the transformation\n$$\n\\bigotimes_{v\\in V}\\ket i_v\\mapsto\\bigotimes_{v\\in V}\\ket i_v\\ket0_I\\ket{\\text{init}_i},$$\nwhere\n$$\\ket{\\text{init}_i}=\\bigotimes_{v\\in V,u\\in S_i}\\left|\\widetilde d^\\ell_{G,w}(u,v)\\right\\rangle_v\\ket{G''_i,w''_i},$$\nand $d^\\ell_{G,w}(u,v)$ is given in Lemma~\\ref{lem:bounded_hop_distance}, $G''_i$ and $w''_i$ are given in lemma~\\ref{lem:approx_distance}.\n\n\\item {\\bf $\\text{Setup}_i$:} Perform the transformation\n$$\\bigotimes_{v\\in V}\\ket i_v\\ket0_I\\ket{\\text{init}_i}\\mapsto\\bigotimes_{v\\in V}\\ket i_v\\left(\\sum_{s\\in S_i}\\frac1{|S_i|}\\ket s_I\\ket{\\text{data}_i(s)}\\right)\\ket{\\text{init}_i},$$\nwhere\n$\\ket{\\text{data}_i(s)}=\\bigotimes_{v\\in V}\\ket s_v\\bigotimes_{v\\in V,u\\in S_i}\\left|\\widetilde d_{G''_i,w''_i}(s,u)\\right\\rangle_v$.\n\n\\item {\\bf $\\text{Evaluation}_i$:} Perform the transformation\n$$\\bigotimes_{v\\in V}\\ket i_v\\left(\\ket{s,0}_I\\ket{\\text{data}_i(s)}\\right)\\ket{\\text{init}_i}\\mapsto\\bigotimes_{v\\in V}\\ket i_v\\left(\\ket{s,f_i(s)}_I\\ket{\\text{data}_i(s)}\\right)\\ket{\\text{init}_i}.$$\n\\end{itemize}\n\nWe now analyze the round complexity:\n\\begin{itemize}\n\\item In $\\widetilde O\\left(D_G+\\frac n{\\varepsilon\\cdot r}+r\\right)$ rounds, each $v\\in V$ can know $\\widetilde d^\\ell_{G,w}(u,v)$ for each $u\\in S_i$, with high probability, due to Lemma~\\ref{lem:bounded_hop_mssp}.\nAfter that, the overlay network $\\left(G''_i,w''_i\\right)$ can be embedded in $O(D_G+rk)$ rounds due to Lemma~\\ref{lem:embedding} (we say that the network $G=(V,E)$ embeds an overlay network $G'=(V',E')$ with a weight function $w':E'\\to\\mathbb N^+$ if $V'\\subseteq V$ and for each $v\\in V'$, it stores each $e\\in E'$ incident to $v$ along with $w'(e)$ in the local memory).\nTherefore, the procedure $\\text{Initialization}_i$ can be implemented in $T_0=\\widetilde O\\left(D_G+\\frac n{\\varepsilon\\cdot r}+rk\\right)$ rounds.\n\\item The node leader can collect $S_i$ in $O(D_G+r)$ rounds.\nIt then prepares the quantum state $\\sum_{s\\in S_i}\\frac1{|S_i|}\\ket s_I$ and broadcasts to all nodes using CNOT copies, in $O(D_G)$ rounds.\nThus, the transformation\n$$\\bigotimes_{v\\in V}\\ket i_v\\ket0_I\\ket{\\text{init}_i}\\mapsto\\bigotimes_{v\\in V}\\ket i_v\\left(\\sum_{s\\in S_i}\\frac1{|S_i|}\\ket s_I\\bigotimes_{v\\in V}\\ket s_v\\right)\\ket{\\text{init}_i}$$\ncan be implemented in $O(D_G+r)$ rounds.\nBesides, the transformation\n$$\\bigotimes_{v\\in V}\\ket i_v\\bigotimes_{v\\in V}\\ket s_v\\ket{\\text{init}_i}\\mapsto\\bigotimes_{v\\in V}\\ket i_v\\bigotimes_{v\\in V}\\ket s_v\\bigotimes_{v\\in V,u\\in S_i}\\left|\\widetilde d^{4|S_i|\/k}_{G''_i,w''_i}(s,u)\\right\\rangle_v\\ket{\\text{init}_i}$$\ncan be implemented in $T_1=\\widetilde O\\left(\\frac r{\\varepsilon\\cdot k}\\cdot D_G+r\\right)$ rounds since Lemma~\\ref{lem:sssp_on_overlay} implies that, after the overlay network $\\left(G''_i,w''_i\\right)$ is embedded, each $v\\in V$ can know $\\widetilde d^{4|S_i|\/k}_{G''_i,w''_i}(s,u)$ for each $u\\in S_i$ within $T_1$ rounds.\nTherefore, the procedure $\\text{Setup}_i$ can be implemented in $T_1=\\widetilde O\\left(\\frac r{\\varepsilon\\cdot k}\\cdot D_G+r\\right)$ rounds.\n\\item For the procedure $\\text{Evaluation}_i$, recall that $f_i(s)=\\max_{v\\in V}\\widetilde d_{G,w,i}(s,v)$ where\n$$\\widetilde d_{G,w,i}(s,v)=\\min_{u\\in S_i}\\left\\{\\widetilde d^{4|S_i|\/k}_{G''_i,w''_i}(s,u)+\\widetilde d^\\ell_{G,w}(u,v)\\right\\}.$$\nBy definition, for any $v\\in V$ and $u\\in S_i$, $\\widetilde d^{4|S_i|\/k}_{G''_i,w''_i}(s,u)$ and $\\widetilde d^\\ell_{G,w}(u,v)$ have been stored in the local memory of $v$, i.e., $\\ket\\cdot_v$.\nThus, each $v\\in V$ can locally compute $\\widetilde d_{G,w,S_i}(s,v)$, and the node leader can compute the maximum by converge-casting in $O(D_G)$ rounds.\nSo the procedure $\\text{Evaluation}_i$ can be implemented in $T_2=O(D_G)$ rounds.\n\\end{itemize}\nBy Lemma~\\ref{lem:optimization}, there exists a quantum procedure maximizing $f_i$ in $\\widetilde O(T_0+\\sqrt r(T_1+T_2))$ rounds with high probability.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:upper_bound}]\nWe give a quantum procedure maximizing $f$ also by following the framework of distributed quantum optimization:\n\\begin{itemize}\n\\item \\textbf{Initialization} is a classical procedure which samples vertex sets $S_1,\\cdots,S_n$, and $\\ket{\\text{init}}$ represents the corresponding classical information.\n\\item \\textbf{Setup:} Perform the transformation\n$$\\ket0_I\\ket{\\text{init}}\\mapsto\\sum_{i=1}^n\\frac1n\\ket i_I\\ket{\\text{data}(i)}\\ket{\\text{init}},$$\nwhere $\\ket{\\text{data}(i)}=\\bigotimes_{v\\in V}\\ket i_v$.\n\\item \\textbf{Evaluation:} Perform the transformation\n$$\\ket{i,0}_I\\ket{\\text{data}(i)}\\ket{\\text{init}}\\mapsto\\ket{i,f(i)}_I\\ket{\\text{data}(i)}\\ket{\\text{init}}.$$\n\\end{itemize}\n\nWe now analyze the round complexity:\n\\begin{itemize}\n\\item $S_1,\\cdots,S_n$ are sampled locally in parallel, and the procedure Initialization is free, i.e., $T_0=0$.\n\\item The node leader prepares the quantum state $\\sum_{i=1}^n\\frac1n\\ket i_I$ and broadcast using CNOT copies to all nodes.\nTherefore, the procedure Setup can be implemented in $T_1=O(D_G)$ rounds.\n\\item The procedure Evaluation can be of $T_2=\\widetilde O\\left(D_G+\\frac n{\\varepsilon\\cdot r}+rk+\\sqrt r\\left(\\frac r{\\varepsilon\\cdot k}\\cdot D_G+r\\right)\\right)$ rounds by Lemma~\\ref{lem:evaluation}.\n\\end{itemize}\n\nBy Lemma~\\ref{lem:optimization} and Lemma~\\ref{lem:approximation}, there exists a quantum procedure that find, with high probability, some $i\\in[1,n]$ such that $D_{G,w}\\le f(i)\\le(1+\\varepsilon)^2D_{G,w}$, in\n$$\\widetilde O(T_0+\\sqrt{n\/r}(T_1+T_2))=\\widetilde O\\left(\\sqrt{n\/r}\\left(D_G+\\frac n{\\varepsilon\\cdot r}+rk+\\sqrt r\\left(\\frac r{\\varepsilon\\cdot k}\\cdot D_G+r\\right)\\right)\\right)$$\nrounds.\nBy the choice of the parameters in Eq.~\\eqref{eqn:alg_parameters}, Theorem~\\ref{thm:upper_bound} follows.\n\\end{proof}\n\\section{Toolkits in Nanongkai's Algorithm}\n\\label{sec:toolkits}\n\nLet $G=(V,E)$ be a distributed network with a weight function $w:E\\to\\mathbb N^+$ and a pre-defined node $\\text{leader}\\in V$.\nWe assume that each node initially knows $n=|V|$ and $W=\\max_{e\\in E}w(e)$.\nThe parameters $\\varepsilon,r,\\ell,k$ are chosen the same as in Eq.~\\eqref{eqn:alg_parameters}.\nWe follow the background of Section~\\ref{sec:approx_ecc}.\nGiven a vertex set $S\\subseteq V$, let $\\widetilde d^\\ell_{G,w}(\\cdot)$, $(G'_S,w'_S)$, $N^k_S(\\cdot)$, $(G''_S,w''_S)$, $\\widetilde d_{G,w,S}(\\cdot)$ be as defined in Lemma~\\ref{lem:bounded_hop_distance} and Lemma~\\ref{lem:approx_distance}.\nThe following lemmas and algorithms are summarized from \\cite[arXiv version]{Nanongkai14STOC}.\n\n\\begin{lemma}[Theorem 3.2 in \\cite{Nanongkai14STOC}]\nFor $s\\in V$ known to all nodes, there exists an algorithm (Algorithm~\\ref{alg:bounded_hop_sssp}) such that in $\\widetilde O(\\ell\/\\varepsilon)$ rounds, each $v\\in V$ knows $\\widetilde d^\\ell_{G,w}(s,v)$, and during the whole computation, each node broadcasts $O(\\log n)$ messages of size $O(\\log n)$ to its neighbors.\n\\label{lem:bounded_hop_sssp}\n\\end{lemma}\n\n\\begin{algorithm}\n\\caption{Bounded-Hop SSSP $(G,w,s,\\ell,\\varepsilon)$}\n\\label{alg:bounded_hop_sssp}\n\\begin{algorithmic}[1]\n\\Require Network $(G,w)$, source node $s$ and parameters $\\ell,\\varepsilon>0$.\n\\Ensure Each node $v$ knows $\\widetilde d^\\ell_{G,w}(s,v)$.\n\\State Initially, $\\widetilde d^\\ell_{G,w}(s,v)\\leftarrow\\infty$ for each $v\\in V$.\n\\For{$i=0$ to $\\log\\frac{2nW}\\varepsilon$}\n\\State Run bounded-distance SSSP with parameters $(G,w_i,s,(1+2\/\\varepsilon)\\ell)$ using Algorithm~\\ref{alg:bounded_distance_sssp}.\n\\For{each $v\\in V$} in parallel\n\\If{$d_{G,w_i}(s,v)\\le(1+2\/\\varepsilon)\\ell$}\n\\State $\\widetilde d^\\ell_{G,w}(s,v)\\leftarrow\\min\\left\\{\\widetilde d^\\ell_{G,w}(s,v),d_{G,w_i}(s,v)\\right\\}$.\n\\EndIf\n\\EndFor\n\\EndFor\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{algorithm}\n\\caption{Bounded-Distance SSSP $(G,w,s,L)$}\n\\label{alg:bounded_distance_sssp}\n\\begin{algorithmic}[1]\n\\Require Network $(G,w)$, source node $s$ and parameter $L>0$.\n\\Ensure Each node $v$ knows whether $d_{G,w}(s,v)\\le L$, and if so, it further knows $d_{G,w}(s,v)$.\n\\State Initially, $d_{G,w}(s,s)\\leftarrow0$ and $d_{G,w}(s,v)\\leftarrow\\infty$ for each $v\\ne s$.\n\\State Let $t$ be the time this algorithm starts.\n\\For{round $r=t$ to $t+L$}\n\\For{each $v\\in V$} in parallel\n\\For{each message $(u,d_{G,w}(s,u))$ received in the previous round}\n\\If{$d_{G,w}(s,u)+w(\\{u,v\\})\\le L$}\n\\State $d_{G,w}(s,v)\\leftarrow\\min\\left\\{d_{G,w}(s,v),d_{G,w}(s,u)+w(\\{u,v\\})\\right\\}$.\n\\EndIf\n\\EndFor\n\\If{$d_{G,w}(s,v)=r-t$}\n\\State $v$ broadcasts message $(v,d_{G,w}(s,v))$ to all neighbors.\n\\EndIf\n\\EndFor\n\\EndFor\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{lemma}[Theorem 3.6 and Lemma 3.7 in \\cite{Nanongkai14STOC}]\nThere exist an algorithm (Algorithm~\\ref{alg:bounded_hop_mssp}) such that in $\\widetilde O(D_G+\\ell\/\\varepsilon+|S|)$ rounds, each node $v\\in V$ knows $\\widetilde d^\\ell_{G,w}(s,v)$ for each $s\\in S$, with probability of failure at most $n^{-c}$, for any constant $c>0$ and sufficiently large $n$.\n\\label{lem:bounded_hop_mssp}\n\\end{lemma}\n\n\\begin{algorithm}[!h]\n\\caption{Bounded-Hop Multi-Source Shortest Paths $(G,w,S,\\ell,\\varepsilon)$}\n\\label{alg:bounded_hop_mssp}\n\\begin{algorithmic}[1]\n\\Require Network $(G,w)$, set of source nodes $S$ and parameters $\\ell,\\varepsilon>0$.\n\\Ensure With high probability, each node $v$ knows $\\widetilde d^\\ell_{G,w}(s,v)$ for each $s\\in S$.\n\\State Assume that $S=\\{s_1,\\cdots,s_b\\}$.\nLet $\\mathcal A_i$ be the Algorithm~\\ref{alg:bounded_hop_sssp} with parameters $(G,w,s_i,\\ell,\\varepsilon)$ for each $i\\in[1,k]$ (each $\\mathcal A_i$ is of $T=\\widetilde O(\\ell\/\\varepsilon)$ rounds, and during the whole computation of $\\mathcal A_i$, each node broadcasts $O(\\log n)$ messages to its neighbors due to Lemma~\\ref{lem:bounded_hop_sssp}).\n\\State The node leader samples $\\Delta_1,\\cdots,\\Delta_b\\in[0,b\\log n]$ independently and uniformly at random for delaying algorithms $\\mathcal A_1,\\cdots,\\mathcal A_k$, and broadcasts them by pipelining in $O(D_G+b)$ rounds.\n\\For{$r=1$ to $T+b\\log n$}\n\\For{each $v\\in V$} in parallel\n\\State Let $a=|\\ \\{i\\in[1,b]:v\\text{ broadcasts a message in the }(r-\\Delta_i)\\text{-th round of }\\mathcal A_i\\}\\ |$.\n\\If{$a\\le\\lceil\\log n\\rceil$}\n\\State $v$ broadcasts these $a$ messages in the next $\\lceil\\log n\\rceil$ rounds.\n\\Else\n\\State The algorithm fails.\n\\EndIf\n\\EndFor\n\\EndFor\n\\end{algorithmic}\n\\end{algorithm}\n\n\\newpage\n\n\\begin{lemma}[Theorem 4.5 in \\cite{Nanongkai14STOC}]\nAfter the overlay network $(G'_S,w'_S)$ is embedded, there exists an algorithm (Algorithm~\\ref{alg:embedding}) which further embeds the overlay network $(G''_S,w''_S)$ in $\\widetilde O(D_G+|S|k)$\nrounds.\n\\label{lem:embedding}\n\\end{lemma}\n\n\\begin{algorithm}\n\\caption{Embedding Overlay Network $(G,w,S,G'_S,w'_S,k)$}\n\\label{alg:embedding}\n\\begin{algorithmic}[1]\n\\Require Network $(G,w)$, set of source nodes $S$, overlay network $(G'_S,w'_S)$ and parameter $k>0$.\n\\Ensure It embeds the overlay network $(G''_S,w''_S)$.\n\\State Each node $s\\in S$ broadcasts the $k$ shortest edges incident to it on $(G'_S,w'_S)$ (this can be done in $O(D_G+|S|k)$ rounds).\n\\For{each $s\\in S$} locally\n\\State $s$ computes $N^k_S(s)$, along with the weight $w''_S(\\{s,v\\})=d_{G'_S,w'_S}(s,v)$ for each $v\\in N^k_S(s)$ (this can be done due to Observation 3.12 in \\cite{Nanongkai14STOC}).\n\\EndFor\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{lemma}[Lemma 4.6 in \\cite{Nanongkai14STOC}]\nFor node $s\\in S$ known to all nodes, after the overlay network $(G''_S,w''_S)$ is embedded, there exists an algorithm (Algorithm~\\ref{alg:sssp_on_overlay}) such that in $\\widetilde O\\left(\\frac{|S|}{\\varepsilon\\cdot k}\\cdot D_G+|S|\\right)$ rounds, each node $v\\in V$ knows for each $u\\in S$ the value of $\\widetilde d^{4|S|\/k}_{G''_S,w''_S}(s,u)$.\n\\label{lem:sssp_on_overlay}\n\\end{lemma}\n\n\\begin{algorithm}\n\\caption{SSSP on Overlay Network $(G,w,S,\\varepsilon,k,G''_S,w''_S,s)$}\n\\label{alg:sssp_on_overlay}\n\\begin{algorithmic}[1]\n\\Require Network $(G,w)$, set of source nodes $S$, parameters $\\varepsilon,k>0$, overlay network $(G''_S,w''_S)$ and source node $s\\in S$.\n\\Ensure Each node $v$ knows $\\widetilde d^{4|S|\/k}_{G''_S,w''_S}(s,u)$ for each $u\\in S$.\n\\State Let $\\mathcal A$ be the Algorithm~\\ref{alg:bounded_hop_sssp} with parameters $(G''_S,w''_S,s,4|S|\/k,\\varepsilon)$ ($\\mathcal A$ is of $T=\\widetilde O\\left(\\frac{|S|}{\\varepsilon\\cdot k}\\right)$ rounds, and during the whole computation of $\\mathcal A$, each node broadcasts $O(\\log n)$ messages to its neighbors due to Lemma~\\ref{lem:bounded_hop_sssp}).\n\\For{$r=1$ to $T$}\n\\State Let $a$ be the number of nodes in $G''_S$ that want to broadcast a message to its neighbors in $G''_S$ in the $r$-th round of $\\mathcal A$.\nCount $a$ and make every nodes in $G$ knows $a$ in $O(D_G)$ rounds.\n\\State Each node in $G''_S$, which wants to send a message to each of its neighbors in $G''_S$, broadcasts such message to all nodes in $G$ (this takes $O(D_G+a)$ rounds).\n\\EndFor\n\\end{algorithmic}\n\\end{algorithm}\n\\section*{Acknowledgements}\nWe thank the anonymous reviewers' feedback.\nThis work was supported in part by, the National Key R\\&D Program of China 2018YFB1003202, National Natural Science Foundation of China (Grant No. 61972191), the Program for Innovative Talents and Entrepreneur in Jiangsu, and Anhui Initiative in Quantum Information Technologies (Grant No. AHY150100).\n\n\\bibliographystyle{acm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and main results}\nThe study of real-valued topological cocycles and real skew product extensions has been initiated by Besicovitch, Gottschalk, and Hedlund.\nBesicovitch \\cite{Be} proved the existence of point transitive real skew product extensions of an irrational rotation on the one-dimensional torus.\nFurthermore, he proved that none of them is minimal, i.e. there are always non-transitive points for a point transitive real skew product extension.\nThe main result in Chapter 14 of \\cite{G-H} can be rephrased to the dichotomy that a topologically conservative real skew product extension of a minimal rotation on a torus (finite or infinite dimensional) is either point transitive or it is defined by a topological coboundary and almost periodic.\nThis result and a generalisation to skew product extensions of a Kronecker transformation (cf. \\cite{LM}) exploit the isometric behaviour of a minimal rotation.\nA corresponding result apart from isometries is based on homotopy conditions for the class of distal minimal homeomorphisms usually called Furstenberg transformations (cf. \\cite{Gr}).\nHowever, in general this dichotomy is not valid, and counterexamples can be provided by the Rokhlin skew products of the so-called topological type $III_0$.\nThis motivates the study of topologically conservative real skew product extensions of compact flows apart from isometries and toral extensions, which is carried out in this note for \\emph{distal} minimal flows with Abelian compactly generated acting groups.\n\nThroughout this note we shall denote by $T$ a \\emph{compactly generated Abelian} Hausdorff topological group acting continuously on a compact metric phase space $(X,d)$ so that $(X,T)$ is a \\emph{compact metric flow}.\nIn the monograph \\cite{G-H} such an acting group $T$ is called \\emph{generative}, and notions of recurrence are provided for such Abelian acting groups apart from $\\mathbb Z$ and $\\mathbb R$.\nFor a $\\mathbb Z$-action on $X$ we let $T$ be the self-homeomorphism of $X$ generating the action by $(n,x)\\mapsto T^n x$, while in the case of a real flow we shall use the notation $\\{\\phi^t:t\\in\\mathbb R\\}$ for the acting group.\nWe call a flow \\emph{minimal} if the whole phase space is the only non-empty invariant closed subset, and then for every $x\\in X$ the \\emph{orbit closure} $\\bar{\\mathcal O}_T(x)=\\overline{\\{\\tau x:\\tau\\in T\\}}$ is all of $X$.\nA flow $(X,T)$ is \\emph{topologically transitive} if for arbitrary open neighbourhoods $\\mathcal U,\\mathcal V\\subset X$ there exists some $\\tau\\in T$ with $\\tau\\mathcal U\\cap\\mathcal V\\neq\\emptyset$, and it is \\emph{weakly mixing} if the flow $(X\\times X,T)$ with the diagonal action is topologically transitive.\nFor a topologically transitive flow $(X,T)$ with complete separable metric phase space there exists a dense $G_\\delta$-set of \\emph{transitive points} $x$ with $\\bar{\\mathcal O}_T(x)=X$, and a flow with transitive points is \\emph{point transitive}.\nIf $(X,T)$ and $(Y,T)$ are flows with the same acting group $T$ and $\\pi:X\\longrightarrow Y$ is a continuous \\emph{onto} mapping with $\\pi(\\tau x)=\\tau\\pi(x)$ for every $\\tau\\in T$ and $x\\in X$, then $(Y,T)=\\pi(X,T)$ is called a factor of $(X,T)$ and $(X,T)$ is called an extension of $(Y,T)$.\nSuch a mapping $\\pi$ is called a \\emph{homomorphism} of the flows $(X,T)$ and $(Y,T)$.\nThe set of bicontinuous bijective homomorphisms of a flow $(X,T)$ onto itself is the topological group $\\textup{Aut}(X,T)$ of \\emph{automorphisms} of $(X,T)$ with the topology of uniform convergence.\nTwo points $x,y\\in X$ are called \\emph{distal} if\n\\begin{equation*}\n\\inf_{\\tau\\in T}d(\\tau x,\\tau y)> 0 ,\n\\end{equation*}\notherwise they are called \\emph{proximal}.\nFor a general compact Hausdorff flow $(X,T)$ distality of two points $x,y\\in X$ is defined by the absence of any nets $\\{\\tau_n\\}_{n\\in I}\\subset T$ with $\\lim \\tau_n x=\\lim \\tau_n y$.\nA flow is called distal if any two distinct points are distal, and an extension of flows is called distal if any two distinct points in the same fibre are distal.\nAn important property of distal compact flows is the \\emph{partitioning} of the phase space into invariant closed minimal subsets, even if the flow is not minimal.\n\nSuppose that $\\mathbb A$ is an Abelian locally compact second countable (Abelian l.c.s.) group with zero element $\\mathbf 0_\\mathbb A$, and let $\\mathbb A_\\infty$ denote its one point compactification with the convention that $g+\\infty=\\infty+g=\\infty$ for every $g\\in \\mathbb A$.\nA cocycle of a compact metric flow $(X,T)$ is a continuous mapping $f:T\\times X\\longrightarrow \\mathbb A$ with the identity\n\\begin{equation*}\nf(\\tau,\\tau' x)+f(\\tau',x)=f(\\tau\\tau',x)\n\\end{equation*}\nfor all $\\tau,\\tau'\\in T$ and $x\\in X$.\nGiven a compact metric $\\mathbb Z$-flow $(X,T)$ and a continuous function $f:X\\longrightarrow \\mathbb A$, we can define a cocycle $f:\\mathbb{Z}\\times X \\longrightarrow \\mathbb A$ with $f(1,\\cdot)\\equiv f$ by\n\\begin{equation*}\nf(n,x)=\n\\begin{cases}\n\\sum_{k=0}^{n-1}f(T^k x) & \\textup{if}\\enspace n\\geq 1 ,\n\\\\\n\\mathbf 0_\\mathbb A & \\textup{if}\\enspace n=0 ,\n\\\\\n-f(-n,T^n x) & \\textup{if}\\enspace n < 0.\n\\end{cases}\n\\end{equation*}\nMoreover, there is a natural occurrence of cocycles of $\\mathbb R$-flows as solutions to ODE's.\nSuppose that $(M,\\{\\phi^t:t\\in\\mathbb R\\})$ is a smooth flow on a compact manifold $M$ and $A:M\\longrightarrow\\mathbb R$ is a smooth function.\nThen a continuous real valued cocycle $f(t,m)$ of the flow $(M,\\{\\phi^t:t\\in\\mathbb R\\})$ is given by the fundamental solution to the ODE\n\\begin{equation*}\n\\frac{d\\, f(t,m)}{d\\, t}=A(\\phi^t(m))\n\\end{equation*}\nwith the initial condition $f(0,m)=0$.\nThe \\emph{skew product extension} of the flow $(X,T)$ by a cocycle $f:T\\times X\\longrightarrow \\mathbb A$ is defined by the homeomorphisms\n\\begin{equation*}\n\\widetilde\\tau_f(x,a)=(\\tau x, f(\\tau,x)+ a)\n\\end{equation*}\nof $X\\times \\mathbb A$ for all $\\tau\\in T$, which provide a continuous action $(\\tau,x,a)\\mapsto\\widetilde\\tau_f(x,a)$ of $T$ on $X\\times \\mathbb A$ by the cocycle identity.\nFor a $\\mathbb Z$-flow $(X,T)$ this action is generated by\n\\begin{equation*}\n\\widetilde T_f(x,a)=(T x, f(x)+ a) .\n\\end{equation*}\nThe essential property of a skew product is that the $T$-action on $X\\times\\mathbb A$ commutes with the right translation action of the group $\\mathbb A$ on $X\\times\\mathbb A$, which is defined by $$R_b(x,a)=(x,a-b)$$\n for every $b\\in\\mathbb A$.\nThe orbit closure of $(x,a)\\in X\\times\\mathbb A$ under $\\widetilde\\tau_f$ will be denoted by $\\bar{\\mathcal O}_{T,f}(x,a)=\\overline{\\{\\widetilde\\tau_f(x,a):\\tau\\in T\\}}$.\n\nThe \\emph{prolongation} $\\mathcal D_T(x)$ of $x\\in X$ under the group action of $T$ is defined by\n\\begin{equation*}\n\\mathcal D_T(x)=\\bigcap\\{\\bar{\\mathcal O}_T(\\mathcal U):\\mathcal U\\enspace\\textup{is an open neighbourhood of}\\enspace x\\},\n\\end{equation*}\nand we shall use the notation $\\mathcal D_{T,f}(x,a)$ for the prolongation of a point $(x,a)\\in X\\times \\mathbb A$ under the skew product action $\\widetilde\\tau_f$.\n\nWhile the inclusion of the orbit closure in the prolongation is obvious, the coincidence of these sets is generic by a result from the paper \\cite{Gl3}.\nThis result, one of our main tools, is usually referred to as ``topological ergodic decomposition''.\n\n\\begin{fact}\\label{fact:o_p}\nFor every compact \\emph{metric} flow $(X,T)$ there exists a $T$-invariant dense $G_\\delta$ set $\\mathcal F\\subset X$ so that for every $x\\in\\mathcal F$ holds\n\\begin{equation*}\n\\bar{\\mathcal O}_{T}(x)=\\mathcal D_{T}(x) .\n\\end{equation*}\nFor a skew product extension $\\widetilde\\tau_f$ of $(X,T)$ by a cocycle $f:T\\times X\\longrightarrow \\mathbb A$ there exists a $T$-invariant dense $G_\\delta$ set $\\mathcal F\\subset X$ so that for every $x\\in\\mathcal F$ and \\emph{every} $a\\in \\mathbb A$ holds\n\\begin{equation*}\n\\bar{\\mathcal O}_{T,f}(x,a)=\\mathcal D_{T,f}(x,a) .\n\\end{equation*}\nThis assertion holds as well for the extension of $\\widetilde\\tau_f$ to $X\\times \\mathbb A_\\infty$ which is defined by $(x,\\infty)\\mapsto (\\tau x,\\infty)$ for every $x\\in X$, and given an $\\mathbb R^2$-valued topological cocycle $g=(g_1,g_2):T\\times X\\longrightarrow\\mathbb R^2$ for the extension of $\\widetilde\\tau_g$ to $X\\times(\\mathbb R_\\infty)^2$ which is defined by $(x,s,\\infty)\\mapsto (\\tau x,s+g_1(x),\\infty)$, $(x,\\infty,t)\\mapsto (\\tau x,\\infty,t+g_2(x))$, and $(x,\\infty,\\infty)\\mapsto (\\tau x,\\infty,\\infty)$, for every $x\\in X$ and $s,t\\in\\mathbb R$.\n\\end{fact}\n\n\\begin{proof}\nThe statement for a compact metric phase space and a general acting group is according to Theorem 1 of \\cite{AkGl}.\nThe other statements can be verified by means of the extension of $\\widetilde\\tau_f$ onto the compactification of $X\\times\\mathbb A$.\nThe coincidence of $\\bar{\\mathcal O}_{T,f}(x,a)$ and $\\mathcal D_{T,f}(x,a)$ for some $(x,a)\\in X\\times \\mathbb A$ implies this coincidence for all $(x,a')\\in\\{x\\}\\times \\mathbb A_\\infty$, since the extension of $\\widetilde\\tau_f$ to $X\\times\\mathbb A_\\infty$ commutes with the right translation on $X\\times \\mathbb A_\\infty$.\n\\end{proof}\n\n\\begin{remark}\\label{rem:o_p}\nIf $y\\in\\bar{\\mathcal O}_{T}(x)$ and $z\\in\\bar{\\mathcal O}_{T}(y)$, then $z\\in\\bar{\\mathcal O}_{T}(x)$ follows by a diagonalisation argument.\nA corresponding statement for prolongations is not valid, however follows from $x\\in\\bar{\\mathcal O}_{T}(y)$ and $z\\in\\mathcal D_{T}(y)$ that $z\\in\\mathcal D_{T}(x)$.\n\\end{remark}\n\nWe shall consider more general Abelian acting groups than $\\mathbb Z$ and $\\mathbb R$, hence the definition of recurrence requires the notions of a replete semigroup and an extensive subset of the Abelian compactly generated group $T$ (cf. \\cite{G-H}).\nWe recall that a semigroup $P\\subset T$ is replete if for every compact subset $K\\subset T$ there exists a $\\tau\\in T$ with $\\tau K\\subset P$, and a subset $E\\subset T$ is extensive if it intersects every replete semigroup.\nTherefore, a subset $E$ of $T=\\mathbb Z$ or $T=\\mathbb R$ is extensive if and only if $E$ contains arbitrarily large positive \\emph{and} arbitrarily large negative elements.\n\n\\begin{defi}\nWe call a cocycle $f(\\tau,x)$ of a minimal compact metric flow $(X,T)$ \\emph{topologically recurrent} if for arbitrary neighbourhoods $\\mathcal U\\subset X$ and $U(\\mathbf 0_\\mathbb A)\\subset \\mathbb A$ of $\\mathbf 0_\\mathbb A$ there exists an extensive set of elements $\\tau\\in T$ with\n\\begin{equation*}\n\\mathcal U\\cap\\tau^{-1}(\\mathcal U)\\cap\\{x\\in X: f(\\tau,x)\\in U(\\mathbf 0_\\mathbb A)\\}\\neq\\emptyset .\n\\end{equation*}\nSince $\\widetilde\\tau_f$ and the right translation on $X\\times\\mathbb A$ commute, this is equivalent to the \\emph{regional recurrence} of the skew product action $\\widetilde\\tau_f$ on $X\\times\\mathbb A$, i.e. for every open neighbourhood $U\\subset X\\times \\mathbb A$ there exists an extensive set of elements $\\tau\\in T$ with $\\widetilde\\tau_f(U)\\cap U\\neq\\emptyset$.\nA non-recurrent cocycle is called \\emph{transient}.\n\nA point $(x,a)\\in X\\times \\mathbb A$ is $\\widetilde\\tau_f$-recurrent if for every neighbourhood $U\\subset X\\times \\mathbb A$ of $(x,a)$ the set of $\\tau\\in T$ with $\\widetilde\\tau_f(x,a)\\in U$ is extensive.\nMoreover, a point $(x,a)\\in X\\times\\mathbb A$ is \\emph{regionally} $\\widetilde\\tau_f$-recurrent if for every neighbourhood $U$ of $(x,a)$ the set of $\\tau\\in T$ with $\\widetilde\\tau_f(U)\\cap U\\neq\\emptyset$ is extensive.\n\\end{defi}\n\n\\begin{remarks}\\label{rems:rec}\nIf $f(\\tau,x)$ is recurrent, then by Theorems 7.15 and 7.16 in \\cite{G-H} there exists a dense $G_\\delta$ set of $\\widetilde\\tau_f$-recurrent points in $X\\times\\mathbb R$.\n\nGiven a regionally $\\widetilde\\tau_f$-recurrent point $(x,a)\\in X\\times\\mathbb A$, every point in $\\{x\\}\\times\\mathbb A$ is regionally $\\widetilde\\tau_f$-recurrent.\nThe minimality of $(X,T)$ and Theorem 7.13 in \\cite{G-H} imply that every point in $X\\times \\mathbb A$ is regionally $\\widetilde\\tau_f$-recurrent, hence $f(\\tau,x)$ is recurrent.\n\nA cocycle $f(n,x)$ of a $\\mathbb Z$-flow is topologically recurrent if and only if $\\widetilde T_f$ is topologically conservative, i.e. for every open neighbourhood $U\\subset X\\times\\mathbb A$ there exists an integer $n\\neq 0$ so that $\\widetilde T_f^n(U)\\cap U\\neq\\emptyset$.\n\\end{remarks}\n\nOne of the most important concepts in the study of cocycles is the essential range, originally introduced in the measure theoretic category by Schmidt \\cite{Sch}.\n\n\\begin{defi}\\label{def:er}\nLet $f(\\tau,x)$ be a cocycle of a minimal compact metric flow $(X,T)$.\nAn element $a\\in \\mathbb A$ is in the set $E(f)$ of \\emph{topological essential values} if for arbitrary neighbourhoods $\\mathcal U\\subset X$ and $U(a)\\subset \\mathbb A$ of $a$ there exists an element $\\tau\\in T$ so that\n\\begin{equation*}\n\\mathcal U\\cap\\tau^{-1}(\\mathcal U)\\cap\\{x\\in X: f(\\tau,x)\\in U(a)\\}\n\\end{equation*}\nis non-empty.\nThe set $E(f)$ is also called the \\emph{topological essential range}.\nThe cocycle identity implies that $f(\\mathbf 1_T,x)=\\mathbf 0_\\mathbb A$ for all $x\\in X$ and hence $\\mathbf 0_\\mathbb A\\in E(f)$.\nMoreover, the essential range is always a closed \\emph{subgroup} of $\\mathbb A$ (cf. \\cite{LM}, Proposition 3.1, which carries over from the case of a minimal $\\mathbb Z$-action to a general Abelian group acting minimally).\n\\end{defi}\n\n\\begin{fact}\\label{fact:er}\nIf $f(\\tau,x)$ is a cocycle with full topological essential range $E(f)=\\mathbb A$, then $\\mathcal D_{T,f}(x,a)\\subset\\{x\\}\\times\\mathbb A$ holds for every $(x,a)\\in X\\times\\mathbb A$.\nBy Fact \\ref{fact:o_p} there exists a $T$-invariant dense $G_\\delta$ set $\\mathcal F\\subset X$ with $\\{x\\}\\times \\mathbb A\\subset\\bar{\\mathcal O}_{T,f}(x,a)$ for every $(x,a)\\in\\mathcal F\\times \\mathbb A$.\nFor every $\\tau\\in T$ follows that $\\{\\tau x\\}\\times \\mathbb A\\subset\\bar{\\mathcal O}_{T,f}(x,g)$, and by the minimality of the flow $(X,T)$ every $(x,a)\\in\\mathcal F\\times \\mathbb A$ is a transitive point for $\\widetilde\\tau_f$.\n\\end{fact}\n\nThroughout this note we shall use a notion of ``relative'' triviality of cocycles.\n\n\\begin{defi}\\label{def:rnt}\nLet $f_1(\\tau,x)$ and $f_2(\\tau,x)$ be $\\mathbb R$-valued cocycles of a minimal compact metric flow $(X,T)$.\nWe shall call the cocycle $f_2(\\tau,x)$ \\emph{relatively trivial} with respect to $f_1(\\tau,x)$, if for every sequence $\\{(\\tau_k,x_k)\\}_{k\\geq 1}\\subset T\\times X$ with $d(x_k, \\tau_k x_k)\\to 0$ and $f_1(\\tau_k,x_k)\\to 0$ it holds also that $f_2(\\tau_k,x_k)\\to 0$ as $k\\to\\infty$.\nFor a sequence $\\{\\tau_k\\}_{k\\geq 1}\\subset T$ and a point $\\bar x\\in X$ so that $\\tau_k\\bar x$ and $f_1(\\tau_k,\\bar x)$ are convergent, this implies that also $f_2(\\tau_k,\\bar x)$ is convergent.\n\\end{defi}\n\nBy the following lemma it suffices to verify an essential value condition ``locally''.\n\n\\begin{lemma}\\label{lem:trans}\nLet $(X,T)$ be a minimal compact metric flow with an \\emph{Abelian} group $T$ acting, and let $f(\\tau,x)$ be a cocycle of $(X,T)$ with values in an Abelian l.c.s. group $\\mathbb A$.\nIf there exists a sequence $\\{(\\tau_k,x_k)\\}_{k\\geq 1}\\subset T\\times X$ with $d(x_k, \\tau_k x_k)\\to 0$ and $f(\\tau_k,x_k)\\to a\\in \\mathbb A_\\infty$ ($\\mathbb R_\\infty\\times\\mathbb R_\\infty$ for $\\mathbb A=\\mathbb R^2$, respectively) as $k\\to\\infty$, then for every $x\\in X$ it holds that $(x,a)\\in\\mathcal D_{T,f}(x,\\mathbf 0_\\mathbb A)$.\nHence if $a\\in \\mathbb A$ is finite, then $a\\in E(f)$.\n\nNow let $g=(g_1,g_2):T\\times X\\longrightarrow\\mathbb R^2$ be a cocycle of the flow $(X,T)$ with a sequence $\\{(\\tau_k,x_k)\\}_{k\\geq 1}\\subset T\\times X$ so that $d(x_k,\\tau_k x_k)\\to 0$, $g_1(\\tau_k,x_k)\\to 0$, and $g_2(\\tau_k,x_k)\\nrightarrow 0$ as $k\\to\\infty$.\nThen there exist a point $\\bar x\\in X$ and a sequence $\\{\\bar\\tau_k\\}_{k\\geq 1}\\subset T$ so that $d(\\bar x,\\bar\\tau_k \\bar x)\\to 0$ and $g(\\bar\\tau_k,\\bar x)\\to(0,\\infty)$ as $k\\to\\infty$.\nMoreover, for an extension $(Y,T)$ of $(X,T)=\\pi(Y,T)$ there exists a sequence $\\{(\\tilde\\tau_k,y_k)\\}_{k\\geq 1}\\subset T\\times Y$ with $d_Y(y_k,\\tilde\\tau_k y_k)\\to 0$ and $(g\\circ\\pi)(\\tilde\\tau_k,y_k)\\to(0,\\infty)$ as $k\\to\\infty$.\n\\end{lemma}\n\n\\begin{proof}\nWe let $\\{(\\tau_k,x_k)\\}_{k\\geq 1}\\subset T\\times X$ be a sequence with the properties above, and we may assume that $x_k\\to x'\\in X$ as $k\\to\\infty$.\nFor arbitrary neighbourhoods $\\mathcal U\\subset X$ and $U(a)$ of $a\\in \\mathbb A_\\infty$ we can fix an element $\\tau\\in T$ with $\\tau x'\\in\\mathcal U$, and since the group $T$ is Abelian it holds that $\\tau x_k\\to\\tau x'$ and $\\tau_k\\tau x_k=\\tau\\tau_k x_k\\to \\tau x'$ as $k\\to\\infty$.\nFrom the cocycle identity and the continuity of $f(\\tau,\\cdot)$ follows\n\\begin{eqnarray*}\nf(\\tau_k,\\tau x_k) & = & f(\\tau,\\tau_k x_k) + f(\\tau_k,x_k)+f(\\tau^{-1},\\tau x_k)\\\\\n& = & f(\\tau,\\tau_k x_k)+ f(\\tau_k,x_k)-f(\\tau, x_k) \\to a\n\\end{eqnarray*}\nas $k\\to\\infty$, and for all $k$ large enough it holds that $\\tau x_k$, $\\tau_k\\tau x_k\\in\\mathcal U$ and $f(\\tau_k,\\tau x_k)\\in U(a)$.\nSince the neighbourhoods $\\mathcal U$ and $U(a)$ were arbitrary, we have $(x,a)\\in\\mathcal D_{T,f}(x,\\mathbf 0_\\mathbb A)$ for every $x\\in X$ and $a\\in E(f)$ if $a\\neq\\infty$.\n\nIf $g(\\tau_k,x_k)\\nrightarrow(0,\\infty)$ as $k\\to\\infty$, then $E(g)$ has an element $(0,c)$ with $c\\in\\mathbb R\\setminus\\{0\\}$.\nSince $E(g)$ is a closed subspace of $\\mathbb R^2$, we can start over with a sequence $\\{(\\tau_k,x_k)\\}_{k\\geq 1}\\subset T\\times X$ so that $d(x_k,\\tau_k x_k)\\to 0$, $g_1(\\tau_k,x_k)\\to 0$, and $|g_2(\\tau_k,x_k)|\\to\\infty$ as $k\\to\\infty$.\nThe statement above implies that $(x,0,\\infty)\\in\\mathcal D_{T,g}(x,0,0)$ for every $x\\in X$, and by Fact \\ref{fact:o_p} we can select $\\bar x\\in X$ and a sequence $\\{\\bar\\tau_k\\}_{k\\geq 1}\\subset T$ so that $\\bar\\tau_k\\bar x\\to\\bar x$ and $g(\\bar\\tau_k,\\bar x)\\to(0,\\infty)$.\nFor an arbitrary point $\\bar y\\in\\pi^{-1}(\\bar x)$ we can select an increasing sequence of positive integers $\\{k_l\\}_{l\\geq 1}$ with $d_Y(\\bar\\tau_{k_{l+1}} \\bar y,\\bar\\tau_{k_l}\\bar y)\\to 0$ and $(g\\circ\\sigma)(\\bar\\tau_{k_{l+1}}(\\bar\\tau_{k_l})^{-1},\\bar\\tau_{k_l}\\bar y)\\to(0,\\infty)$ and put $\\{(\\tilde\\tau_l,y_l)=(\\bar\\tau_{k_{l+1}}(\\bar\\tau_{k_l})^{-1},\\bar\\tau_{k_l}\\bar y)\\}_{l\\geq 1}$.\n\\end{proof}\n\n\\begin{defi}\nLet $f(\\tau,x)$ be a cocycle of a minimal compact metric flow $(X,T)$ with values in an Abelian l.c.s. group $\\mathbb A$, and let $b:X\\longrightarrow \\mathbb A$ be a continuous function.\nAnother cocycle of the flow $(X,T)$ can be defined by the $\\mathbb A$-valued function\n\\begin{equation*}\ng(\\tau,x)=f(\\tau,x)+b(\\tau x)-b(x).\n\\end{equation*}\nThe cocycle $g(\\tau,x)$ is called \\emph{topologically cohomologous} to the cocycle $f(\\tau,x)$ with the \\emph{transfer function} $b(x)$.\nA cocycle $g(\\tau,x)=b(\\tau x)-b(x)$ topologically cohomologous to zero is bounded on $T\\times X$ and called a \\emph{topological coboundary}.\n\\end{defi}\n\nThe Gottschalk-Hedlund theorem (\\cite{G-H}, Theorem 14.11) characterises topological coboundaries of a minimal $\\mathbb Z$-action as cocycles bounded on at least one semi-orbit.\nThe generalisation to an Abelian group $T$ acting minimally is natural.\n\n\\begin{fact}\\label{fact:GH}\nA real valued topological cocycle $f(\\tau,x)$ of a minimal compact metric flow $(X,T)$ with an Abelian acting group $T$ is a coboundary if and only if there exists a point $\\bar x\\in X$ so that the function $\\tau\\mapsto f(\\tau,\\bar x)$ is bounded on $T$.\nFor the groups $T=\\mathbb Z$ and $T=\\mathbb R$ acting, the boundedness on a semi-orbit is sufficient.\n\nA real valued cocycle $f(\\tau,x)$ is also a topological coboundary if for every sequence $\\{(\\tau_k,x_k)\\}_{k\\geq 1}\\subset T\\times X$ with $d(x_k,\\tau_k x_k)\\to 0$ the set $\\{f(\\tau_k,x_k)\\}_{k\\geq 1}\\subset\\mathbb R$ is bounded.\n\\end{fact}\n\n\\begin{proof}\nSuppose that $\\tau\\mapsto f(\\tau,\\bar x)$ is bounded on $T$.\nBy the cocycle identity holds\n\\begin{equation*}\nf(\\tau,\\tau'\\bar x)=f(\\tau\\tau',\\bar x)-f(\\tau',\\bar x)\n\\end{equation*}\nfor all $\\tau,\\tau'\\in T$, and by the density of the $T$-orbit of $\\bar x$ follows the boundedness of $f(\\tau,x)$ on $T\\times X$ and thus the triviality of the subgroup $E(f)=\\{0\\}$.\nBy the density of the $T$-orbit of $\\bar x$ and the boundedness of $\\tau\\mapsto f(\\tau,\\bar x)$, the intersection $\\{x\\}\\times\\mathbb R\\cap\\bar{\\mathcal O}_{T,f}(\\bar x,0)$ is non-empty for every $x\\in X$.\nFor every $x\\in X$ this intersection is a singleton, since otherwise Lemma \\ref{lem:trans} proves a non-zero element in $E(f)$.\nHence the compact set $\\bar{\\mathcal O}_{T,f}(\\bar x,0)$ is the graph of a continuous function $b:X\\longrightarrow\\mathbb R$ with $f(\\tau,\\bar x)=b(\\tau\\bar x)$, and thus $f(\\tau,x)=b(\\tau x)-b(x)$ holds for every $(\\tau,x)\\in T\\times X$.\nFor $T=\\mathbb Z$ and $T=\\mathbb R$ the set of \\emph{limit points} of a semi-orbit is a $T$-invariant closed subset of $X$, which is non-empty by compactness and equal to $X$ by minimality.\nWe can conclude the proof as above, but using the semi-orbit.\n\nNow suppose that $f(\\tau,x)$ is not a topological coboundary and let $\\bar x\\in X$ be arbitrary.\nThen there exists a sequence $\\{\\tau'_l\\}_{l\\geq 1}\\subset T$ with $|f(\\tau'_l,\\bar x)|\\to\\infty$, and we may assume that $\\tau'_l \\bar x\\to x'$ as $l\\to\\infty$.\nSince $(X,T)$ is minimal, there exists sequence $\\{\\tau''_k\\}_{k\\geq 1}\\subset T$ with $\\tau''_k x'\\to\\bar x$ as $k\\to\\infty$.\nA diagonalisation with a sufficiently increasing sequence of positive integers $\\{l_k\\}_{l\\geq 1}$ yields for $\\tau_k=\\tau_k''\\tau_{l_k}'$ that $\\tau_k\\bar x\\to\\bar x$ and $|f(\\tau_k,\\bar x)|=|f(\\tau_k'',\\tau_{l_k}'\\bar x)+f(\\tau_{l_k}',\\bar x)|\\to\\infty$ as $k\\to\\infty$.\n\\end{proof}\n\nThe following lemma appeared originally in the paper \\cite{A} in a setting for $\\mathbb R^d$-valued cocycles of a minimal rotation on a torus.\n\n\\begin{lemma}\\label{lem:at}\nLet $f(\\tau,x)$ be a real valued topological cocycle of a minimal compact metric flow $(X,T)$ with an Abelian acting group $T$.\nIf the skew product action $\\widetilde\\tau_f$ is \\emph{not} point transitive on $X\\times\\mathbb R$, then for every neighbourhood $U\\subset\\mathbb R$ of $0$ there exist a compact symmetric neighbourhood $K\\subset U$ of $0$ and an $\\varepsilon>0$ so that for every $\\tau\\in T$ holds\n\\begin{equation}\\label{eq:s_l}\n\\{x\\in X:d(x,\\tau x)<\\varepsilon\\enspace\\textup{and}\\enspace f(\\tau,x)\\in 2K\\setminus K^0\\}=\\emptyset .\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $f(\\tau,x)$ is real valued and $\\widetilde\\tau_f$ is not point transitive.\nBy Fact \\ref{fact:er} the essential range $E(f)$ is a proper closed subgroup of $\\mathbb R$, and thus there exists a compact symmetric neighbourhood $K\\subset U$ of $0$ with $(2K\\setminus K^0)\\cap E(f)=\\emptyset$.\nIf the assertion is false for the neighbourhood $K$, then there exists a sequence $\\{(\\tau_k,x_k)\\}_{k\\geq 1}\\subset T\\times X$ with $d(x_k, \\tau_k x_k)\\to 0$ and $f(\\tau_k,x_k)\\to t\\in 2K\\setminus K^0$.\nNow Lemma \\ref{lem:trans} implies $t\\in E(f)\\cap 2K\\setminus K^0$, in contradiction to the choice of $K$.\n\\end{proof}\n\nWe shall commence the study of cocycles of distal minimal flows by the generalisation of the results for minimal rotations in \\cite{G-H} and \\cite{LM}.\n\n\\begin{proposition}\\label{prop:isom}\nLet $(X,T)$ be a minimal compact \\emph{isometric} flow with a compactly generated Abelian acting group $T$, and let $f(\\tau,x)$ be a topologically recurrent real valued cocycle of $(X,T)$.\nThen the cocycle $f(\\tau,x)$ is either a coboundary or its skew product extension $\\widetilde \\tau_f$ is point transitive on $X\\times\\mathbb R$.\n\\end{proposition}\n\n\\begin{proof}\nSuppose that the cocycle $f(\\tau,x)$ is not a coboundary and $\\widetilde \\tau_f$ is not point transitive.\nThen by Lemma \\ref{lem:at} there exist a compact symmetric neighbourhood $K$ of $0$ and an $\\varepsilon>0$ so that equality (\\ref{eq:s_l}) holds for every $\\tau\\in T$.\nFurthermore, if $L\\subset T$ is a compact generative subset, then $\\varepsilon>0$ can be chosen small enough so that for all $\\tau'\\in L$ and $x,x'\\in X$ with $d(x,x')<\\varepsilon$ it holds that\n\\begin{equation*}\nf(\\tau',x)-f(\\tau',x')\\in K^0 .\n\\end{equation*}\nBy Fact \\ref{fact:GH} we can fix a pair $(\\bar\\tau,\\bar x)\\in T\\times X$ with $d(\\bar x,\\bar\\tau\\bar x)<\\varepsilon$ and $f(\\bar\\tau,\\bar x)\\notin 2K$, since $f(\\tau,x)$ is not a coboundary.\nThe Abelian group $T$ acts on $X$ isometrically, and thus $d(\\bar x,\\bar\\tau\\bar x)<\\varepsilon$ implies that $d(\\tau'\\bar x,\\bar\\tau\\tau' \\bar x)=d(\\tau'\\bar x,\\tau'\\bar\\tau\\bar x)<\\varepsilon$.\nTogether with equality (\\ref{eq:s_l}) we can conclude for every $\\tau'\\in L$ that\n\\begin{equation*}\nf(\\bar\\tau,\\tau'\\bar x)=f(\\bar\\tau,\\bar x)-f(\\tau',\\bar x)+f(\\tau',\\bar\\tau\\bar x)\\notin 2K ,\n\\end{equation*}\nand hence both of the real numbers $f(\\bar\\tau,\\bar x)$ and $f(\\bar\\tau,\\tau'\\bar x)$ are elements of the one and the same of the disjoint sets $\\mathbb R^+\\setminus 2K$ and $\\mathbb R^-\\setminus 2K$.\nSince the set $L$ is generative in the Abelian group $T$ acting minimally on $X$, it follows by induction that $f(\\bar\\tau, x)$ is in the closure of one of the sets $\\mathbb R^+\\setminus 2K$ and $\\mathbb R^-\\setminus 2K$ for every $x\\in X$.\nThus we have a constant $c>0$ with $|f(\\bar\\tau^k,x)|> |k| c$ for every integer $k$, and we define a subset $P\\subset T$ by\n\\begin{equation*}\nP=\\cup_{k\\geq 1}\\bar\\tau^k\\cdot\\{\\tau\\in T:f(\\tau,\\cdot)<|k| c\/2\\} .\n\\end{equation*}\nGiven two integers $k,k'\\geq 1$ and $\\bar\\tau^k\\tau$, $\\bar\\tau^{k'}\\tau'\\in P$ with $f(\\tau,\\cdot)<|k| c\/2$ and $f(\\tau',\\cdot)<|k'| c\/2$, we can conclude that $\\bar\\tau^k\\tau\\bar\\tau^{k'}\\tau'=\\bar\\tau^{k+k'}(\\tau\\tau')$ with $f(\\tau\\tau',\\cdot)<|k+k'| c\/2$, hence $P$ is a semigroup.\nMoreover, the semigroup $P$ contains a translate of every compact set $L\\subset T$, since for large enough $k\\geq 1$ the inequality $f(\\tau,x)<|k| c\/2$ holds for every $\\tau\\in L$ and every $x\\in X$.\nTherefore $P$ is a replete semigroup in $T$ so that $|f(\\tau,x)|>c\/2$ holds for every $(\\tau,x)\\in P\\times X$, which contradicts the existence of a dense $G_\\delta$ set of $\\widetilde\\tau_f$-recurrent points (cf. Remarks \\ref{rems:rec}).\n\\end{proof}\n\nThe \\emph{Rokhlin extensions} and the \\emph{Rokhlin skew products} have been studied in the measure theoretic setting in \\cite{LL} and \\cite{LP}.\nWe shall introduce the notion of a \\emph{perturbed Rokhlin skew product}, which will be inevitable in our main result.\n\n\\begin{defi}\nSuppose that $(X,T)$ is a distal minimal compact metric flow and $(M,\\{\\phi^t:t\\in\\mathbb R\\})$ is a distal minimal compact metric $\\mathbb R$-flow.\nLet $f:T\\times X\\longrightarrow\\mathbb R$ be a cocycle of $(X,T)$ with a point transitive skew product $\\widetilde\\tau_f$ on $X\\times\\mathbb R$.\nWe define the \\emph{Rokhlin extension} $\\tau_{\\phi,f}$ on $X\\times M$ by\n\\begin{equation*}\n\\tau_{\\phi,f}(x,m)=(\\tau x,\\phi^{f(\\tau,x)}(m)) ,\n\\end{equation*}\nwhich is an action of the group $T$ on $X\\times M$ due to the cocycle identity for $f(\\tau,x)$.\nIf $(\\bar x,0)$ is a transitive point for $\\widetilde\\tau_f$, then by the minimality of $(M,\\{\\phi^t:t\\in\\mathbb R\\})$ every point $(\\bar x,m)$ with $m\\in M$ is a transitive point for $(X\\times M,T)$.\nSince the flow $(X\\times M,T)$ is distal by the distality of its components, it is even minimal.\n\nThe skew product extension of $(X\\times M,T)$ by the cocycle $(\\tau,x,m)\\mapsto f(\\tau,x)$ is the \\emph{Rokhlin skew product} $\\widetilde\\tau_{\\phi,f}$ on $X\\times M\\times \\mathbb R$ with\n\\begin{equation*}\n\\widetilde\\tau_{\\phi,f}(x,m,t)=(\\tau x,\\phi^{f(\\tau,x)}(m),t+f(\\tau,x)) .\n\\end{equation*}\nLet $g:\\mathbb R\\times M\\longrightarrow\\mathbb R$ be a cocycle of the flow $(M,\\{\\phi^t:t\\in\\mathbb R\\})$.\nThe $\\mathbb R$-valued map\n\\begin{equation*}\n(\\tau,x,m)\\mapsto f(\\tau,x)+g(f(\\tau,x),m)\n\\end{equation*}\ndefined on $T\\times X\\times M$ turns out to be a cocycle of the flow $(X\\times M,T)$ due to the cocycle identity for $g(t,m)$.\nThe skew product extension of this cocycle with\n\\begin{equation*}\n\\widetilde\\tau_{\\phi,f,g}(x,m,t)=(\\tau x,\\phi^{f(\\tau,x)}(m),t+f(\\tau,x)+g(f(\\tau,x),m)) .\n\\end{equation*}\nis called a \\emph{perturbed Rokhlin skew product} $\\widetilde\\tau_{\\phi,f,g}$ on $X\\times M\\times\\mathbb R$.\n\\end{defi}\n\nWe shall present at first the basic example of a topological Rokhlin skew product of topological type $III_0$, i.e. recurrent with a trivial topological essential range but not a topological coboundary.\n\n\\begin{example}\\label{ex:zi}\nLet $f:\\mathbb T\\longrightarrow\\mathbb R$ be a continuous function with a point transitive skew product extension $\\widetilde T_f$ of the irrational rotation $T$ by $\\alpha$ on the torus, and let $\\beta\\in(0,1)$ be irrational so that the $\\mathbb R$-flow $(\\mathbb T^2,\\{\\phi^t:t\\in\\mathbb R\\})$ defined by $\\phi^t(y,z)=(y+t,z+\\beta t)$ is minimal and distal.\nThe minimal and distal Rokhlin extension $T_{\\phi,f}$ on $\\mathbb T^3$ is\n\\begin{equation*}\nT_{\\phi,f}(x,y,z)=(x+\\alpha,y+f(x),z+\\beta f(x)),\n\\end{equation*}\nand putting $h(x,y,z)=f(x)$ for all $(x,y,z)\\in\\mathbb T^3$ gives a topological type $III_0$ cocycle $h(n,(x,y,z))$ of the homeomorphism $T_{\\phi,f}$ with the skew product extension $\\widetilde T_{\\phi,f}$.\nIndeed, since $\\widetilde T_f$ is point transitive, the cocycle $h(n,(x,y,z))$ is recurrent, but it is not bounded and therefore no topological coboundary.\nFurthermore, a sequence $\\{t_n\\}_{n\\geq 1}\\subset\\mathbb R$ with $t_n\\mod 1\\to 0$ and $(\\beta t_n)\\mod 1\\to 0$ cannot have a finite cluster point apart from zero, and hence $E(h)=\\{0\\}$.\nFor a point $\\bar x\\in\\mathbb T$ so that $(\\bar x,0)\\in\\mathbb T\\times\\mathbb R$ is transitive under $\\widetilde T_f$ and arbitrary $y,z\\in\\mathbb T$ the orbit closure of $(\\bar x,y,z,0)$ under the skew product extension of $T_{\\phi,f}$ by $h$ is of the form\n\\begin{equation*}\n\\bar{\\mathcal O}_{\\widetilde T_{\\phi,f}}((\\bar x,y,z),0)=\\bar{\\mathcal O}_{T_{\\phi,f},h}((\\bar x,y,z),0)=\\mathbb T\\times\\{(\\phi^t(y,z),t)\\in\\mathbb T^2\\times\\mathbb R:t\\in\\mathbb R\\} .\n\\end{equation*}\nThe collection of these sets is a partition of $\\mathbb T^3\\times\\mathbb R$ into $\\widetilde T_{\\phi,f}$-orbit closures.\n\\end{example}\n\nThe next example makes clear that the perturbation of a Rokhlin skew product by a cocycle is an essential component, which in general cannot be eliminated by continuous cohomology.\n\n\\begin{example}\\label{ex:pe}\nLet $T$, $f$, $h$, and $\\{\\phi^t:t\\in\\mathbb R\\}$ be defined as in Example \\ref{ex:zi}, and suppose that $g(t,(y,z))$ is a point transitive $\\mathbb R$-valued cocycle of the flow $\\{\\phi^t:t\\in\\mathbb R\\}$.\nFrom the unique ergodicity of the flow $\\{\\phi^t:t\\in\\mathbb R\\}$ follows $\\int_{\\mathbb T^2} g(t,(y,z)) d\\lambda(y,z)=0$ for every $t\\in\\mathbb R$, and after rescaling $g$ we can assume that $|g(t,(y,z))|<|t|\/2$ for every $t\\in\\mathbb R$ and $(y,z)\\in\\mathbb T^2$.\nWe define a function\n\\begin{equation*}\n\\bar h(x,y,z)= f(x)+g(f(x),(y,z))\n\\end{equation*}\nso that the cocycle of $T_{\\phi,f}$ is $\\bar h(n,(x,y,z))= f(n,x)+g(f(n,x),(y,z))$ for every $n$.\nSince the perturbation $g(f(n,x),(y,z))$ is unbounded, there cannot be a continuous transfer function defined on $\\mathbb T^3$ so that $\\bar h$ and $h$ are cohomologous.\nHowever, due to the condition $|g(t,(y,z))|<|t|\/2$ the set\n\\begin{equation*}\n\\mathbb T\\times\\{(\\phi^t(y,z),t+g(t,(y,z)))\\in\\mathbb T^2\\times\\mathbb R:t\\in\\mathbb R\\}\n\\end{equation*}\nis closed, and it coincides with $\\bar{\\mathcal O}_{\\widetilde T_{\\phi,f,g}}((\\bar x,y,z),0)$ if the point $(\\bar x,0)\\in\\mathbb T\\times\\mathbb R$ is transitive under $\\widetilde T_f$.\nThus the structure of the orbit closures is preserved as well as these sets provide a partition of $\\mathbb T^3\\times\\mathbb R$.\n\\end{example}\n\n\\begin{remark}\nThe structure of Example \\ref{ex:zi} can be revealed from the toral extensions of $T_{\\phi,f}$ by the function $(\\gamma h) \\mod 1$ for all $\\gamma\\in\\mathbb R$.\nThis distal homeomorphism of $\\mathbb T^4$ is transitive and hence minimal for rationally independent $1$, $\\beta$, and $\\gamma$.\nHowever, for $\\gamma=1$ and $\\gamma=\\beta$ the orbit closures collapse to graphs representing the dependence of $h$ and the action on the coordinates of the torus.\nThe same approach will not be successful with respect to Example \\ref{ex:pe}.\nIt can be verified that for every $\\gamma\\in\\mathbb R$ the toral extension of $T_{\\phi,f}$ by the function $(\\gamma\\bar h) \\mod 1$ is minimal on $\\mathbb T^4$.\n\\end{remark}\n\nThe main result of this note puts these examples into a structure theorem.\n\n\\begin{strth*}\nSuppose that $(X,T)$ is a distal minimal compact metric flow with a compactly generated Abelian acting group $T$ and that $f:T\\times X\\longrightarrow\\mathbb R$ is a topologically recurrent cocycle which is not a coboundary.\nThen there exist a factor $(X_\\alpha,T)=\\pi_\\alpha(X,T)$, a topological cocycle $f_\\alpha:T\\times X_\\alpha\\longrightarrow\\mathbb R$ of $(X_\\alpha,T)$, and a distal minimal compact metric $\\mathbb R$-flow $(M,\\{\\phi^t:t\\in\\mathbb R\\})$, so that the Rokhlin extension $(X_\\alpha\\times M,T)$ with the action $\\tau_{\\phi,f_\\alpha}$ is a factor $(Y,T)=\\pi_Y(X,T)$ of $(X,T)$.\nThe cocycle $f(\\tau,x)$ is topologically cohomologous to $(f_Y\\circ\\pi_Y)(\\tau,x)=f(\\tau,x)+b(\\tau x)-b(x)$ with a topological cocycle $f_Y:T\\times Y\\longrightarrow\\mathbb R$ of the flow $(Y,T)$ so that\n\\begin{equation}\\label{eq:fpy}\n\\mathcal D_{T, f_Y\\circ\\pi_Y}(x,0)\\cap(\\pi_\\alpha^{-1}(\\pi_\\alpha(x))\\times\\{0\\})=\\pi_Y^{-1}(\\pi_Y(x))\\times\\{0\\}\n\\end{equation}\nholds for all $x\\in X$.\nMoreover, there exists a cocycle $g:\\mathbb R\\times M\\longrightarrow\\mathbb R$ of the $\\mathbb R$-flow $(M,\\{\\phi^t:t\\in\\mathbb R\\})$ so that the cocycle $(\\mathbbm 1+g)(t,m)= t+g(t,m)$ is topologically \\emph{transient} and\n\\begin{equation}\\label{eq:f_Y2}\nf_Y(\\tau,(x,m))=f_\\alpha(\\tau,x)+g(f_\\alpha(\\tau,x),m)=(\\mathbbm 1+g)(f_\\alpha(\\tau,x),m)\n\\end{equation}\nholds for every $\\tau\\in T$ and $(x,m)\\in Y=X_\\alpha\\times M$.\nThus the skew product $\\widetilde\\tau_{f_Y}$ on $Y\\times\\mathbb R$ is the perturbed Rokhlin skew product $\\widetilde \\tau_{\\phi,f_\\alpha, g}$.\n\\end{strth*}\n\n\\noindent We shall conclude the proof of this theorem in the next section of this note.\n\\medskip\n\nThe application of the structure theorem for a topological ergodic decomposition requires a suitable topology on the hyperspace of the non-compact space $X\\times\\mathbb R$.\nWe shall use the Fell topology on the hyperspace of \\emph{non-empty} closed subsets of a locally compact separable metric space.\nGiven finitely many open neighbourhoods $\\mathcal U_1,\\dots,\\mathcal U_k$ and a compact set $K$, an element of the Fell topology base consists of all non-empty closed subsets which intersect each of the open neighbourhoods $\\mathcal U_1,\\dots,\\mathcal U_k$ while being disjoint from $K$.\nThis topology is separable, metrisable, and $\\sigma$-compact (cf. \\cite{HLP}).\nThe Fell topology was introduced in \\cite{Fe} as a compact topology on the hyperspace of all closed subsets, with the empty set as infinity.\n\n\\begin{decth*}\nSuppose that $f:T\\times X\\longrightarrow\\mathbb R$ is a topologically recurrent cocycle of a distal minimal compact metric flow $(X,T)$ with a compactly generated Abelian acting group $T$.\nThe prolongations $\\mathcal D_{T,f}(x,s)\\subset X\\times\\mathbb R$ of the skew product action $\\widetilde\\tau_{f}$ with $(x,s)\\in X\\times\\mathbb R$ define a \\emph{partition} of $X\\times\\mathbb R$.\nThe mapping $(x,s)\\mapsto\\mathcal D_{T,f}(x,s)$ is continuous with respect to the Fell topology on the hyperspace of non-empty closed subsets of $X\\times\\mathbb R$, and the right translation on $X\\times\\mathbb R$ is a minimal continuous $\\mathbb R$-action on the set of prolongations.\nIf the cocycle $f(\\tau,x)$ is not a topological coboundary, then the set of all prolongations in the skew product is Fell compact.\n\\end{decth*}\n\n\\begin{tmath*}\nA recurrent cocycle $f(\\tau,x)$ apart from a coboundary has a minimal compact metric flow as a topological version of the \\emph{Mackey action}.\nIts phase space is the set of prolongations in the skew product with the Fell topology, with the right translation of $\\mathbb R$ acting on the prolongations.\nThis flow is a distal extension (possibly the trivial extension) of a weakly mixing compact metric flow (possibly the trivial flow).\nThe Mackey action is distal if and only if the perturbation cocycle $g(t,m)$ in the structure theorem is a topological coboundary.\n\\end{tmath*}\n\nWhile most of the properties of the topological Mackey action are part of the decomposition theorem, its structure as a distal extension of a weakly mixing flow will be verified in the next section of this note.\nThe proof of the decomposition theorem depends on the following general lemma on transient cocycles of minimal $\\mathbb R$-flows, which might be of independent interest.\n\n\\begin{lemma}\\label{lem:tr_coc}\nLet $(M,\\{\\phi^t:t\\in\\mathbb R\\})$ be a minimal compact metric $\\mathbb R$-flow and let $h(t,m):\\mathbb R\\times M\\longrightarrow\\mathbb R$ be a \\emph{transient} cocycle of $(M,\\{\\phi^t:t\\in\\mathbb R\\})$.\nThen for every point $(m,s)\\in M\\times\\mathbb R$ the orbit $\\mathcal O_{\\phi,h}(m,s)$, the orbit closure $\\bar{\\mathcal O}_{\\phi,h}(m,s)$, and the prolongation $\\mathcal D_{\\phi,h}(m,s)$ under the skew product extension $\\widetilde{\\phi^t}_h$ coincide.\nThe mapping from points to their orbits in $M\\times\\mathbb R$ is continuous with a compact range with respect to the Fell topology, and the right translation on $M\\times\\mathbb R$ provides a minimal continuous $\\mathbb R$-action on the set of orbits.\nMoreover, for every $m\\in M$ the mapping $t\\mapsto h(t,m)$ maps $\\mathbb R$ \\emph{onto} $\\mathbb R$.\n\\end{lemma}\n\n\\begin{proof}\nSince prolongations are closed sets, it suffices to verify that for every $(m,s)\\in M\\times\\mathbb R$ the orbit and the prolongation coincide.\nOtherwise, there exist two points $(m,s), (m',s')\\in M\\times\\mathbb R$ so that $(m',s')$ is not in the $\\widetilde{\\phi^t}_h$-orbit of $(m,s)$, however there exists a sequence $\\{(t_k,m_k)\\}_{k\\geq 1}\\subset\\mathbb R\\times M$ so that $(t_k,m_k)\\to (+\\infty,m)$ and\n\\begin{equation*}\n\\widetilde{\\phi^{t_k}}_h(m_k,s)=(\\phi^{t_k}(m_k),s+h(t_k,m_k))\\to (m',s') .\n\\end{equation*}\nIf there exists a compact set $L\\subset\\mathbb R$ with $h([0,t_k],m_k)\\subset L$ for all $k\\geq 1$, then $h([0,\\infty),m)\\subset L$ since $m_k\\to m$, and by Fact \\ref{fact:GH} the cocycle $h(t,m)$ is a coboundary in contradiction to its transience.\nTherefore we have an increasing sequence of integers $\\{k_l\\}_{l\\geq 1}$, a sequence $\\{t'_l\\}_{l\\geq 1}\\subset\\mathbb R$ with $t'_l\\in[0,t_{k_l}]$, and $S\\in\\{+1,-1\\}$ so that\n\\begin{equation*}\nS\\cdot h(t'_l,m_{k_l})=\\max_{t\\in[0,t_{k_l}]}S\\cdot h(t,m_{k_l})\\to +\\infty\n\\end{equation*}\nas $l\\to\\infty$.\nFor every limit point $\\bar m$ of the sequence $\\{\\phi^{t'_l}(m_{k_l})\\}_{l\\geq 1}$ it holds that $S\\cdot h(t,\\bar m)\\leq 0$ for all $t\\in\\mathbb R$, and the mapping $t\\mapsto h(t,\\bar m)$ maps each of the sets $\\mathbb R^+$ and $\\mathbb R^-$ onto $S\\cdot \\mathbb R^-$.\nHence for every $t\\in\\mathbb R^+$ there exists a $t'\\in\\mathbb R^-$ with $h(t,\\bar m)=h(t',\\bar m)$, and by the density of the semi-orbit $\\{\\phi^t(\\bar m):t\\in\\mathbb R^+\\}$ (cf. the proof of Fact \\ref{fact:GH}) and the cocycle identity the open set\n\\begin{equation*}\nM_k=\\{m\\in M:|h(t,m)|<2^{-k}\\enspace\\textup{for some}\\enspace t<-k\\}\n\\end{equation*}\nis dense for every integer $k\\geq 1$.\nFor a point $m_k$ in the dense $G_\\delta$ set $\\bigcap_{t\\in\\mathbb Q}\\phi^t(M_k)$, we can find rational numbers $t_1,\\dots,t_k<-k$ so that $\\phi^{t_1+\\cdots+t_l}(m_k)\\in M_k$ and $|h(t_1+\\dots+t_l,m_k)|<k 2^{-k}$ for all $1\\leq l\\leq k$.\nSince $M$ is compact, there exists a point $\\tilde m\\in M$ each neighbourhood of which contains at least two distinct points out of the finite sequence $\\phi^{t_1}(m_k),\\dots,\\phi^{t_1+\\cdots+t_k}(m_k)\\in M_k$ for an infinite set of integers $k\\geq 1$.\nThus $(\\tilde m,0)$ is a regionally recurrent point for the skew product $\\widetilde{\\phi^t}_h$, in contradiction to the Remarks \\ref{rems:rec} and the transience of the cocycle $h(t,m)$.\n\nSuppose for some $m\\in M$ the mapping $t\\mapsto h(t,m)$ is not onto $\\mathbb R$, then either there exists a compact set $L\\subset\\mathbb R$ so that one of the inclusions $h([0,\\infty),m)\\subset L$, $h((-\\infty,0],m)\\subset L$ holds, or there exists a point $\\bar m\\in M$ as above.\nIn each of these cases, we can obtain a contradiction to the transience of $h(t,m)$ as above.\n\nLet $(m,s)\\in M\\times\\mathbb R$ be arbitrary, and let a Fell neighbourhood of its \\emph{closed} $\\widetilde{\\phi^t}_h$-orbit be defined by the open neighbourhoods $\\mathcal U_1,\\dots,\\mathcal U_k$ and a compact set $K$.\nObviously the $\\widetilde{\\phi^t}_h$-orbit of every point in a suitable neighbourhood of $(m,s)$ intersects each of the neighbourhoods $\\mathcal U_1,\\dots,\\mathcal U_k$.\nMoreover, the $\\widetilde{\\phi^t}_h$-orbit is disjoint from $K$ for every point in a suitable neighbourhood of $(m,s)$, since otherwise the $\\widetilde{\\phi^t}_h$-prolongation of $(m,s)$ intersects the compact set $K$.\nThis contradicts the coincidence of orbits and prolongations of $\\widetilde{\\phi^t}_h$, and hence the mapping of a point $(m,s)\\in M\\times\\mathbb R$ to its $\\widetilde{\\phi^t}_h$-orbit is Fell continuous.\nBy the surjectivity of the mapping $t\\mapsto h(t,m)$, every $\\widetilde{\\phi^t}_h$-orbit intersects the set $\\{(m,0):m\\in M\\}$.\nThe Fell compactness of the set of $\\widetilde{\\phi^t}_h$-orbits in $M\\times\\mathbb R$ follows, and the right translation on $M\\times\\mathbb R$ is a Fell continuous $\\mathbb R$-action by the definition of the Fell neighbourhoods.\nSince the points $(\\phi^t(m),0)$ and $(m,-h(m,t))$ are within the same $\\widetilde{\\phi^t}_h$-orbit, the minimality of $(M,\\{\\phi^t:t\\in\\mathbb R\\})$ implies the minimality of the right translation action.\n\\end{proof}\n\n\\begin{proof}[Proof of the decomposition theorem]\nFor a topological coboundary $f(\\tau,x)$ the assertions are trivial, since the prolongations are just the right translates of the graph of the transfer function.\nIf the cocycle $f(\\tau,x)$ is not a coboundary, then we apply the structure theorem.\nFor a point $\\bar x_\\alpha\\in X_\\alpha$ with $\\bar{\\mathcal O}_{T,f_\\alpha}(\\bar x_\\alpha,0)=X_\\alpha\\times\\mathbb R$ and every $m\\in M$ follows from equality (\\ref{eq:f_Y2}) that $X_\\alpha\\times\\mathcal O_{\\phi,\\mathbbm 1+g}(m,0)\\subset\\bar{\\mathcal O}_{T,f_Y}(\\bar x_\\alpha,m,0)\\subset X_\\alpha\\times\\bar{\\mathcal O}_{\\phi,\\mathbbm 1+g}(m,0)$, and  by Lemma \\ref{lem:tr_coc} holds $\\mathcal O_{\\phi,\\mathbbm 1+g}(m,0)=\\bar{\\mathcal O}_{\\phi,\\mathbbm 1+g}(m,0)=\\mathcal D_{\\phi,\\mathbbm 1+g}(m,0)$.\nThe set of points $\\bar x_\\alpha\\in X_\\alpha$ with $\\bar{\\mathcal O}_{T,f_\\alpha}(\\bar x_\\alpha,0)=X_\\alpha\\times\\mathbb R$ is a dense $G_\\delta$ set, and for arbitrary $(x_\\alpha,m)\\in X_\\alpha\\times M$ and $(x_\\alpha',m',t')\\in\\mathcal D_{T,f_Y}(x_\\alpha,m,0)$ we can find a sequence $\\{(\\tau_k,\\bar x_k,m_k)\\}\\subset T\\times X_\\alpha\\times M$ with $(\\bar x_k,m_k)\\to (x_\\alpha,m)$, $\\tau_k(\\bar x_k,m_k)\\to (x_\\alpha',m')$, $f_Y(\\tau_k,(\\bar x_k, m_k))\\to t'$, and $X_\\alpha\\times\\mathcal O_{\\phi,\\mathbbm 1+g}(m_k,0)=\\bar{\\mathcal O}_{T,f_Y}(\\bar x_k,m_k,0)$.\nTherefore $(m',t')\\in\\mathcal D_{\\phi,\\mathbbm 1+g}(m,0)$ so that $\\mathcal D_{T,f_Y}(x_\\alpha,m,0)\\subset X_\\alpha\\times\\mathcal D_{\\phi,\\mathbbm 1+g}(m,0)$.\nWe conclude for every $(x_\\alpha,m,s)\\in X_\\alpha\\times M\\times\\mathbb R$ that\n\\begin{equation*}\n\\mathcal D_{T,f_Y}(x_\\alpha,m,s)=X_\\alpha\\times\\{(\\phi^t(m),s+t+g(t,m)):t\\in\\mathbb R\\} ,\n\\end{equation*}\nand these sets define a partition of $X_\\alpha\\times M\\times\\mathbb R$.\nThe Fell continuity of the mapping $(x_\\alpha,m,s)\\mapsto\\mathcal D_{T,f_Y}(x_\\alpha,m,s)$, the compactness of its range, and the minimality of the right translation follow directly from Lemma \\ref{lem:tr_coc}.\n\nSuppose that $(x,s)\\in X\\times\\mathbb R$ and $(x_\\alpha',m',s')\\in\\mathcal D_{T,f_Y}(\\pi_Y(x),s)$, and let $\\{\\tau_k\\}_{k\\geq 1}\\subset T$ and $\\{y_k\\}_{k\\geq 1}\\subset X_\\alpha\\times M$ be sequences with $y_k\\to\\pi_Y(x)$, $\\tau_k y_k\\to (x_\\alpha',m')$, and $f_Y(\\tau_k,y_k)\\to s'-s$.\nSince $\\pi_Y$ is a distal homomorphism, it is an open onto mapping, and the mapping $y\\mapsto\\pi_Y^{-1}(y)$ is continuous with respect to the Hausdorff metric $d_H$ (cf. \\cite{K}, p. 68, Theorem 1, and p. 47).\nTherefore we can define a sequence $\\{x_k\\in\\pi_Y^{-1}(y_k)\\}_{k\\geq 1}\\subset X$ so that $x_k\\to x$, $\\tau_k x_k\\to x'\\in\\pi_Y^{-1}(x'_\\alpha,m')$, and $(f_Y\\circ\\pi_Y)(\\tau_k,x_k)\\to s'-s$.\nBy equality (\\ref{eq:fpy}), for every $x''\\in\\pi_Y^{-1}(\\pi_Y(x'))$ there exists a sequence $\\{(\\bar\\tau_k,\\bar x_k)\\}_{k\\geq 1}$ with $\\bar x_k$ sufficiently close to $x_k$ so that $\\bar x_k\\to x$, $\\tau_k\\bar x_k\\to x'$, $(f_Y\\circ\\pi_Y)(\\tau_k,\\bar x_k)\\to s'-s$, $d(\\bar\\tau_k \\tau_k\\bar x_k,x'')<d_H(\\pi_Y^{-1}(\\pi_Y(\\tau_k x_k)),\\pi_Y^{-1}(\\pi_Y(x')))+2^{-k}$, and $(f_Y\\circ\\pi_Y)(\\bar\\tau_k,\\tau_k\\bar x_k)<2^{-k}$.\nFrom $d_H(\\pi_Y^{-1}(\\pi_Y(\\tau_k x_k)),\\pi_Y^{-1}(\\pi_Y(x')))\\to 0$ follows that $\\bar\\tau_k\\tau_k\\bar x_k\\to x''$ and $(f_Y\\circ\\pi_Y)(\\bar\\tau_k\\tau_k,\\bar x_k)\\to s'-s$, hence $(x'',s')\\in\\mathcal D_{T,f_Y\\circ\\pi_Y}(x,s)$.\nThus the $\\widetilde\\tau_{f_Y\\circ\\pi_Y}$-prolongations in $X\\times\\mathbb R$ are exactly the pre-images of the $\\widetilde\\tau_{f_Y}$-prolongations under the mapping $\\pi_Y\\times\\textup{id}_\\mathbb R$.\nIf a Fell neighbourhood of $\\mathcal D_{T,f_Y\\circ\\pi_Y}(x,s)$ is defined by the open neighbourhoods $\\mathcal U_1,\\dots,\\mathcal U_k$ and a compact set $K$ in $X\\times\\mathbb R$, then by the openness of $\\pi_Y$ the $\\pi_Y\\times\\textup{id}_\\mathbb R$-images of these sets define a Fell neighbourhood of $\\mathcal D_{T,f_Y}(\\pi_Y(x),s)$.\nThe Fell continuity of $(x,s)\\mapsto \\mathcal D_{T,f_Y\\circ\\pi_Y}(x,s)$, the compactness of the range, and the minimality of the right translation follow.\n\nLet $b:X\\longrightarrow\\mathbb R$ be the transfer function in the structure theorem.\nThe homeomorphism $(x,s)\\mapsto (x,s+b(x))$ on $X\\times\\mathbb R$ defines a homeomorphism of the hyperspace with the Fell topology commuting with the right translation.\nSince it maps $\\mathcal D_{T,f}(x,s)$ onto $\\mathcal D_{T,f_Y\\circ\\pi_Y}(x,s+b(x))$, it is a topological isomorphism of the right translation actions and all properties carry over.\n\\end{proof}\n\n\\begin{remarks}\nThough the compact metric flow $(X,T)$ is not necessarily itself a Rokhlin extension, the existence of a real-valued recurrent non-coboundary cocycle $f(\\tau,x)$ with a non-transitive skew product extension $\\widetilde\\tau_f$ implies the existence of a Rokhlin extension factor $(Y,T)=\\pi_Y(X,T)$ with a non-trivial flow $\\{\\phi^t:t\\in\\mathbb R\\}$ and a cocycle $f_Y(\\tau,y)$ of the flow $(Y,T)$ with $f_Y\\circ\\pi_Y$ cohomologous to $f$.\n\nThe surjectivity of the mapping $t\\mapsto (\\mathbbm 1+g)(t,m)$ for every $m\\in M$ implies that $\\{f(\\tau,x):\\tau\\in T\\}$ is dense in $\\mathbb R$ for all $x$ in a dense $G_\\delta$ subset of $X$.\n\nFor other well-known topologies on the hyperspace like the Vietoris topology and the Hausdorff topology with respect to the product metric on $X\\times\\mathbb R$, the continuity of the mapping $(x,s)\\mapsto\\mathcal D_{T,f}(x,s)$ can be disproved by Example \\ref{ex:pe}.\n\\end{remarks}\n\n\\section{Proof of the structure theorem}\n\nFurstenberg's structure theorem for distal minimal flows shall be our main tool for studying the structure of cocycles.\n\n\\begin{defi}\nLet $X$ and $Y$ be compact metric spaces, let $\\pi$ be a continuous mapping from of $X$ onto $Y$, and let $M$ be a compact homogeneous metric space.\nSuppose that $\\rho(x_1,x_2)$ is a continuous real valued function defined on the set\n\\begin{equation*}\nR_\\pi=\\{(x,x')\\in X^2: \\pi(x)=\\pi(x')\\}\n\\end{equation*}\nso that for every $y\\in Y$ the function $\\rho$ is a metric on the fibre $\\pi^{-1}(y)$ with an isometric mapping between $\\pi^{-1}(y)$ and $M$.\nThen $X$ is called an $M$-\\emph{bundle} over $Y$.\n\nNow let $(X,T)$ and $(Y,T)=\\pi(X,T)$ be compact metric flows with $X$ an $M$-bundle over $Y$.\nIf the function $\\rho$ satisfies $\\rho(x,x')=\\rho(\\tau x,\\tau x')$ for all $(x,x')\\in R_\\pi$ and $\\tau\\in T$, then $(X,T)$ is called an \\emph{isometric extension} of $(Y,T)$.\n\\end{defi}\n\n\\begin{fact}[Furstenberg's structure theorem \\cite{Fu}]\nLet $(X,T)$ be a distal minimal compact metric flow.\nThen there exist a countable ordinal $\\eta$ and factors $(X_\\xi,T)=\\pi_\\xi(X,T)$ for each ordinal $0\\leq\\xi\\leq\\eta$ with the following properties:\n\\begin{enumerate}\n\\item $(X_\\eta,T)=(X,T)$ and $(X_0,T)$ is the trivial flow.\n\\item $(X_\\xi,T)=\\pi_\\xi^{\\zeta}(X_{\\zeta},T)$ is a factor of $(X_{\\zeta},T)$ for all ordinals $0\\leq\\xi<\\zeta\\leq\\eta$.\n\\item For every ordinal $0\\leq\\xi<\\eta$ the flow $(X_{\\xi+1},T)$ is an isometric extension of $(X_\\xi,T_\\xi)$.\n\\item For a limit ordinal $0<\\xi\\leq\\eta$ the flow $(X_\\xi,T)$ is the inverse limit of the flows $\\{(X_{\\zeta},T):0\\leq\\zeta<\\xi\\}$.\n\\end{enumerate}\nA system $\\{(X_\\xi,T):0\\leq\\xi\\leq\\eta\\}$ with the properties above is called a quasi-isometric system or \\emph{I}-system.\n\\end{fact}\n\n\\begin{defi}\nAn \\emph{I}-system $\\{(X_\\xi,T):0\\leq\\xi\\leq\\eta\\}$ is called \\emph{normal} if for each ordinal $0\\leq\\xi<\\eta$ the flow $(X_{\\xi+1},T)$ is the maximal isometric extension of $(X_{\\xi},T)$ in $(X_{\\eta},T)$.\nThis \\emph{I}-system gives the minimal ordinal $\\eta$ to represent the compact metric flow $(X,T)=(X_\\eta,T)$ (cf. \\cite{Fu}, Proposition 13.1, Definitions 13.2 and 13.3).\n\\end{defi}\n\nIt will be essential that the fibres of all the isometric extensions are connected, with a possible exception of extension from the trivial flow to the minimal isometric flow $(X_1,T)$.\nFor a \\emph{normal} \\emph{I}-system this property will be ensured by results of the paper \\cite{MMWu}.\nThese results require the acting group to be generated by every open neighbourhood of a compact subset, and the group $T$ is even compactly generated.\n\n\\begin{proposition}\nLet $\\{(X_\\xi,T):0\\leq\\xi\\leq\\eta\\}$ be a normal \\emph{I}-system with a compactly generated group $T$ acting minimally and distally.\nThen for all ordinals $1\\leq\\xi<\\zeta\\leq\\eta$ the extension from $(X_\\xi,T)$ to $(X_\\zeta,T)$ has connected fibres.\n\\end{proposition}\n\n\\begin{proof}\nIn the following arguments we shall refer to terminology and results provided in the paper \\cite{MMWu}.\nAt first suppose that $1\\leq\\xi<\\eta$ is not a limit ordinal.\nLet $S(\\pi_{\\xi-1})\\subset X^2$ denote the relativised equicontinuous structure relation of the homomorphism $\\pi_{\\xi-1}:(X,T)\\longrightarrow(X_{\\xi-1},T)$, hence the flow $(X,T)\/S(\\pi_{\\xi-1})$ is the maximal isometric extension $(X_\\xi,T)$ of the flow $(X_{\\xi-1},T)$ in $(X,T)$.\nBy Theorem 3.7 of \\cite{MMWu} the homomorphism $\\pi_\\xi:(X,T)\\longrightarrow (X_\\xi,T)$ has connected fibres.\nFor a limit ordinal $1<\\xi<\\eta$ and an ordinal $0\\leq\\zeta<\\xi$, the same argument shows the connectedness of the fibres of $\\pi_{\\zeta+1}:(X,T)\\longrightarrow(X_{\\zeta+1},T)$.\nSince\n\\begin{equation*}\n(\\pi_\\xi)^{-1}(x_\\xi)=\\bigcap_{0\\leq\\zeta<\\xi}(\\pi_{\\zeta+1})^{-1}(\\pi_{\\zeta+1}^\\xi(x_\\xi))\n\\end{equation*}\nholds for every $x_\\xi\\in X_\\xi$, the fibre $(\\pi_\\xi)^{-1}(x_\\xi)$ is connected as the limit of a sequence of connected sets in a compact metric space (cf. \\cite{K}, p. 170, Theorem 14).\nThe hypothesis follows, since for all ordinals $1\\leq\\xi<\\zeta\\leq\\eta$ the fibres of the homomorphism $\\pi_\\xi^\\zeta$ are the images under $\\pi_\\zeta$ of the connected fibres of $\\pi_\\xi$.\n\\end{proof}\n\nWe shall henceforth assume that $\\{(X_\\xi,T):0\\leq\\xi\\leq\\eta\\}$ is a normal \\emph{I}-system with $(X_\\eta,T)=(X,T)$.\nFor every ordinal $1\\leq\\xi<\\eta$ we shall define a projection of the cocycles of $(X,T)$ to the cocycles of $(X_\\xi,T)$ by families of probability measures, using the fact that every distal extension of compact metric flows is a so-called \\emph{RIM}-extension (relatively invariant measure, cf. \\cite{Gl}).\nFor an isometric extension this \\emph{RIM} is unique (cf. \\cite{Gl}), and within the \\emph{I}-system the \\emph{RIM's} obey to an integral decomposition formula.\n\n\\begin{fact}\nFor every ordinal $0\\leq\\xi\\leq\\eta$ there exists a family of probability measures $\\{\\mu_{\\xi,x_\\xi}:x_\\xi\\in X_\\xi\\}$ on $X$ so that for every $x_\\xi\\in X_\\xi$ and $\\tau\\in T$ holds\n\\begin{equation*}\n\\mu_{\\xi,x_\\xi}(\\pi_\\xi^{-1}(x_\\xi))=1\\enspace\\textup{and}\\enspace\\mu_{\\xi,x_\\xi}\\circ \\tau^{-1}=\\mu_{\\xi, \\tau x_\\xi}.\n\\end{equation*}\nThe mapping $x_\\xi\\mapsto\\mu_{\\xi,x_\\xi}$ is continuous with respect to the weak-* topology on $C(X)^*$.\nFor a continuous function $\\varphi\\in C(X)$ and ordinals $1\\leq\\xi<\\zeta\\leq\\eta$ we have for all $x_\\xi\\in X_\\xi$ the equality that\n\\begin{equation}\n\\mu_{\\xi,x_\\xi}(\\varphi)=\\int_{X_\\zeta}\\mu_{\\zeta,y_\\zeta}(\\varphi)\\, d(\\mu_{\\xi,x_\\xi}\\circ\\pi_\\zeta^{-1})(y_\\zeta) .\n\\end{equation}\n\\end{fact}\n\n\\begin{proof}\nThe proof follows the inductive construction of an invariant measure for $(X,T)$ out of the unique \\emph{RIM's} of the extensions $(X_\\xi,T)=\\pi_\\xi^{\\xi+1}(X_{\\xi+1},T)$ with $0\\leq\\xi<\\eta$ (cf. Chapter 12 of \\cite{Fu}).\nThe generalisation to a relatively invariant measure is provided in \\cite{dVr}, p. 494, where the induction process is initiated with the family of point measures on $X_\\xi$ instead of the point measure on the trivial space $X_0$.\nSince the \\emph{I}-system and the \\emph{RIM's} for the isometric extensions remain fixed, the decomposition formula follows from the inductive construction.\n\\end{proof}\n\nGiven a cocycle $f(\\tau,x)$ of the flow $(X,T)$ and an ordinal $1\\leq\\xi<\\eta$, the \\emph{RIM} $\\{\\mu_{\\xi,x_\\xi}:x_\\xi\\in X_\\xi\\}$ defines a continuous function $f_\\xi:T\\times X_\\xi\\longrightarrow\\mathbb R$ by\n\\begin{equation*}\nf_\\xi(\\tau,x_\\xi)=\\mu_{\\xi,x_\\xi}(f(\\tau,\\cdot))=\\int_X f(\\tau,x)\\,d\\mu_{\\xi,x_\\xi}(x) .\n\\end{equation*}\nThe properties of the \\emph{RIM} imply for all $\\tau,\\tau'\\in T$ and $x_\\xi\\in X_\\xi$ that\n\\begin{eqnarray*}\nf_\\xi(\\tau,\\tau'x_\\xi)+f_\\xi(\\tau',x_\\xi)=\\mu_{\\xi,\\tau'x_\\xi}(f(\\tau,\\cdot))+\\mu_{\\xi,x_\\xi}(f(\\tau',\\cdot))=\\\\\n=\\mu_{\\xi,x_\\xi}(f(\\tau,\\cdot)\\circ\\tau')+\\mu_{\\xi,x_\\xi}(f(\\tau',\\cdot))=f_\\xi(\\tau\\tau',x_\\xi),\n\\end{eqnarray*}\ntherefore $f_\\xi$ is a cocycle of the flow $(X_\\xi,T)$.\nFurthermore, for ordinals $\\xi$, $\\zeta$ with $1\\leq\\xi<\\zeta\\leq\\eta$ and every $x_\\xi\\in X_\\xi$, the integral of the cocycle $(f_\\zeta-f_\\xi\\circ\\pi_\\xi^\\zeta)(\\tau,x_\\zeta)$ by the measure $\\mu_{\\xi,x_\\xi}\\circ\\pi_\\zeta^{-1}$ on $X_\\zeta$ vanishes for every $\\tau\\in T$:\n\\begin{eqnarray*}\n\\int_{X_\\xi}(f_\\zeta-f_\\xi\\circ\\pi_\\xi^\\zeta)(\\tau,x_\\zeta)\\, d(\\mu_{\\xi,x_\\xi}\\circ\\pi_\\zeta^{-1})(x_\\zeta) = \\hspace{3cm}\\\\\n= \\int_{X_\\xi}\\left( \\mu_{\\zeta,x_\\zeta}(f(\\tau,\\cdot))\\right)\\, d(\\mu_{\\xi,x_\\xi}\\circ\\pi_\\zeta^{-1})(x_\\zeta) - \\mu_{\\xi,\\pi_\\xi^\\zeta(x_\\zeta)} (f(\\tau,\\cdot)) = 0\n\\end{eqnarray*}\nSince the measure $\\mu_{\\xi,x_\\xi}\\circ\\pi_\\zeta^{-1}$ is supported by the connected fibre $(\\pi_\\xi^\\zeta)^{-1}(x_\\xi)$ in $X_\\zeta$, for every $\\tau\\in T$ and every $x_\\xi\\in X_\\xi$ the function $x_\\zeta\\mapsto (f_\\zeta-f_\\xi\\circ\\pi_\\xi^\\zeta)(\\tau,x_\\zeta)$ assumes zero on the fibre $(\\pi_\\xi^\\zeta)^{-1}(x_\\xi)$.\nThis property will be essential, as well as the representation of extensions of distal flows by so-called regular extensions.\n\n\\begin{fact}\\label{fact:reg_ex}\nLet $(X,T)$ be a distal minimal compact metric flow with a factor $(Y,T)=\\sigma(X,T)$.\nThen there exist a distal minimal compact Hausdorff flow $(\\tilde X,T)$ with $(X,T)=\\pi(\\tilde X,T)$ as a factor and a Hausdorff topological group $G\\subset\\textup{Aut}(\\tilde X,T)$ acting freely on $\\tilde X$ (i.e. $g(\\tilde x)=\\tilde x$ for some $\\tilde x\\in\\tilde X$ implies $g=\\textbf 1_G$).\nThe group $G$ acts strictly transitive on the fibres $\\tilde\\sigma^{-1}(\\tilde\\sigma(\\tilde x))=\\{g(\\tilde x):g\\in G\\}$ of the homomorphism $\\tilde\\sigma=\\sigma\\circ\\pi$ for every $\\tilde x\\in\\tilde X$, and $(X,T)$ is the orbit space of a subgroup $H$ of $G$ in $\\tilde X$ so that $\\pi$ is the mapping of a point in $\\tilde X$ to its $H$-orbit (cf. \\cite{El} 12.12, 12.13, and 14.26, with a direct proof in \\cite{MMWu}, Proposition 1.1).\n\nFor an isometric extension $(X,T)$ of $(Y,T)$, the flow $(\\tilde X,T)$ is metric and an isometric extension of $(Y,T)$, with a \\emph{compact metric} group $G$ and a compact subgroup $H$, hence called a compact metric group extension.\n\\end{fact}\n\n\\begin{remark}\\label{rem:d_n_m}\nThe construction above is also called the regulariser of an extension.\nIn \\cite{Gl2} it is verified that a compact Hausdorff flow $(\\tilde X,T)$ with these properties is metrisable if and only if the extension from $(Y,T)$ to $(X,T)$ is isometric.\n\\end{remark}\n\nStudying the skew product extensions $\\widetilde\\tau_{f_\\xi}:X_\\xi\\times\\mathbb R\\longrightarrow X_\\xi\\times\\mathbb R$ for the ordinals $0\\leq\\xi\\leq\\eta$ will require the following technical lemma.\n\n\\begin{lemma}\\label{lem:sub}\nLet the minimal compact metric flow $(Z,T)$ be an extension of the flow $(Y,T)=\\sigma(Z,T)$ and let $g(\\tau,y)$ be a real-valued cocycle of $(Y,T)$.\nLet $h(\\tau,z)$ be a real-valued cocycle of $(Z,T)$ so that for every $\\tau\\in T$ and $z'\\in Z$ the image of $\\sigma^{-1}(\\sigma(z'))$ under the function $z\\mapsto h(\\tau,z)$ is connected and includes zero.\nSuppose that there exist a compact symmetric neighbourhood $K\\subset\\mathbb R$ of $0$ and $\\varepsilon>0$ so that for all $z\\in Z$ and $\\tau\\in T$ with $d_Z(z,\\tau z)<\\varepsilon$ holds $(g\\circ\\sigma+h)(\\tau,z)\\notin 2K\\setminus K^0$.\nSuppose that there exists a $\\delta>0$ so that for all $\\tau\\in T$ and $z\\in Z$ with $d_Z(z,\\tau z)<\\delta$ holds $d_Z(z',\\tau z')<\\varepsilon$ for every $z'\\in\\sigma^{-1}(\\sigma(z))$.\nGiven $\\bar z\\in Z$ and a sequence $\\{\\bar\\tau_k\\}\\subset T$ so that $\\bar\\tau_k \\bar z$ converges and $(g\\circ\\sigma)(\\bar\\tau_k,\\bar z)\\to 0$ as $k\\to\\infty$, the sequence $\\{(g\\circ\\sigma+h)(\\bar\\tau_k,\\bar z)\\}_{k\\geq 1}$ is bounded.\nSimilarly, for a sequence $\\{\\bar\\tau_k\\}\\subset T$ so that $\\bar\\tau_k\\bar z$ converges and $(g\\circ\\sigma+h)(\\bar\\tau_k,\\bar z)\\to 0$, the sequence $\\{(g\\circ\\sigma)(\\bar\\tau_k,\\bar z)\\}_{k\\geq 1}$ is bounded.\n\\end{lemma}\n\n\\begin{proof}\nThere exists a $k_0\\geq 1$ so that for all $k,k'\\geq k_0$ holds $d_Z(\\bar\\tau_k\\bar z,\\bar\\tau_{k'}\\bar z)<\\delta$ and\n\\begin{equation*}\n(g\\circ\\sigma)(\\bar\\tau_{k'},\\bar z)-(g\\circ\\sigma)(\\bar\\tau_k,\\bar z)=(g\\circ\\sigma)(\\bar\\tau_{k'}\\bar\\tau_{k}^{-1},\\bar\\tau_k \\bar z)\\in K^0 .\n\\end{equation*}\nBy the choice of $K$, $\\varepsilon$, and $\\delta$ follows that $(g\\circ\\sigma+h)(\\bar\\tau_{k'}\\bar\\tau_k^{-1},z)\\notin 2K\\setminus K^0$ for all $z\\in\\sigma^{-1}(\\sigma(\\bar\\tau_k\\bar z))$.\nSince the range of $(g\\circ\\sigma+h)(\\bar\\tau_{k'}\\bar\\tau_k^{-1},z)$ on the fibre $\\sigma^{-1}(\\sigma(\\bar\\tau_k\\bar z))$ is connected and intersects $K^0$, we can conclude that $(g\\circ\\sigma+h)(\\bar\\tau_{k'}\\bar\\tau_k^{-1},\\bar\\tau_k \\bar z)\\in K^0$ for all $k,k'\\geq k_0$.\nTherefore the sequence $\\{(g\\circ\\sigma+h)(\\bar\\tau_k,\\bar z)\\}_{k\\geq 1}$ is bounded.\n\nProvided a sequence $\\{\\bar\\tau_k\\}\\subset T$ with convergent $\\bar\\tau_k\\bar z$ and $(g\\circ\\sigma+h)(\\bar\\tau_k,\\bar z)\\to 0$, there exists an integer $k_0\\geq 1$ so that for all $k,k'\\geq k_0$ holds $d_Z(\\bar\\tau_k\\bar z,\\bar\\tau_{k'}\\bar z)<\\delta$ and $(g\\circ\\sigma+h)(\\bar\\tau_{k'}\\bar\\tau_{k}^{-1},\\bar\\tau_k \\bar z)\\in K^0$.\nWe conclude as above that $(g\\circ\\sigma+h)(\\bar\\tau_{k'}\\bar\\tau_k^{-1},z)\\in K^0$ for all $k,k'\\geq k_0$ and $z\\in\\sigma^{-1}(\\sigma(\\bar\\tau_k\\bar z))$.\nSince $h(\\bar\\tau_{k'}\\bar\\tau_k^{-1},z)=0$ for some $z\\in\\sigma^{-1}(\\sigma(\\bar\\tau_k\\bar z))$, the sequence $\\{(g\\circ\\sigma)(\\bar\\tau_k,\\bar z)\\}_{k\\geq 1}$ is bounded.\n\\end{proof}\n\nAt first the step from an ordinal to its successor by an isometric extension shall be considered.\nThe ``local'' behaviour within the fibres of a compact group extension is similar to a skew product extension by a compact metric group, even if the global structure might be different since it does not necessarily split into a product.\n\n\\begin{lemma}\\label{lem:c_t}\nLet $\\gamma$ be an ordinal with $1\\leq\\gamma<\\eta$.\nIf there exists a sequence $\\{(\\tau_k,x_k)\\}_{k\\geq 1}\\subset T\\times X_{\\gamma+1}$ with $d_{\\gamma+1} (x_k,\\tau_k x_k)\\to 0$ so that $(f_\\gamma\\circ\\pi_\\gamma^{\\gamma+1})(\\tau_k,x_k)\\to 0$ and $f_{\\gamma+1}(\\tau_k,x_k)\\nrightarrow 0$ as $k\\to\\infty$ (or equivalently $(f_{\\gamma+1}\\circ\\pi_\\gamma^{\\gamma+1})(\\tau_k,x_k)\\nrightarrow 0$ and $f_\\gamma(\\tau_k,x_k)\\to 0$), then the skew product $\\widetilde\\tau_{f_{\\gamma+1}}$ is necessarily point transitive.\nTherefore, if $f_\\gamma(\\tau,x_\\gamma)$ is transient, then $f_{\\gamma+1}(\\tau,x_{\\gamma+1})$ is either transient or the skew product $\\widetilde\\tau_{f_{\\gamma+1}}$ is point transitive.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $\\widetilde\\tau_{f_{\\gamma+1}}$ is not point transitive and let $G\\subset\\textup{Aut}(Z,T)$ define a compact metric group extension of $(X_\\gamma, T)$ with $(X_{\\gamma+1},T)=\\pi(Z,T)$.\nThen the skew product extension $\\widetilde\\tau_{f_{\\gamma+1}\\circ\\pi}$ of the flow $(Z,T)$ is also not point transitive, and Lemma \\ref{lem:at} provides $K\\subset\\mathbb R$ and $\\varepsilon>0$.\nSince $G$ acts uniformly equicontinuous, there exists a $\\delta>0$ so that for all $z\\in Z$ and $\\tau\\in T$ with $d_Z (z,\\tau z)<\\delta$ follows $d_Z(k(z),k(\\tau z))=d_Z(k(z),\\tau k(z))<\\varepsilon$ for all $k\\in K$.\nFor every $z\\in Z$ the $G$-orbit of $z$ is all of the fibre $(\\pi_\\gamma^{\\gamma+1}\\circ\\pi)^{-1}((\\pi_\\gamma^{\\gamma+1}\\circ\\pi)(z))$.\nSince the $\\pi^{\\gamma+1}_\\gamma$-fibres are connected, for every $\\tau\\in T$ and $z'\\in Z$ the range of $(f_{\\gamma+1}-f_\\gamma\\circ\\pi^{\\gamma+1}_\\gamma)(\\tau,\\pi(z))$ on the fibre $(\\pi_\\gamma^{\\gamma+1}\\circ\\pi)^{-1}((\\pi_\\gamma^{\\gamma+1}\\circ\\pi)(z'))$ is connected and contains zero.\nHence Lemma \\ref{lem:sub} applies with $(Y,T)=(X_\\gamma,T)$, $\\sigma=\\pi_\\gamma^{\\gamma+1}\\circ\\pi$, $g=f_\\gamma$, and $h(\\tau,z)=(f_{\\gamma+1}-f_\\gamma\\circ\\pi^{\\gamma+1}_\\gamma)(\\tau,\\pi(z))$.\n\nHowever, given the sequence $\\{(\\tau_k,x_k)\\}_{k\\geq 1}\\subset T\\times X_{\\gamma+1}$ in the hypothesis, Lemma \\ref{lem:trans} provides a point $\\bar x\\in X_{\\gamma+1}$ and a sequence $\\{\\bar\\tau_k\\}\\subset T$ so that $(f_\\gamma\\circ\\pi_\\gamma^{\\gamma+1})(\\bar\\tau_k,\\bar x)\\to 0$ and $f_{\\gamma+1}(\\bar\\tau_k,\\bar x)\\to\\infty$ (or $(f_\\gamma\\circ\\pi_\\gamma^{\\gamma+1})(\\bar\\tau_k,\\bar x)\\to\\infty$ and $f_{\\gamma+1}(\\bar\\tau_k,\\bar x)\\to 0$).\nBy choosing a point $\\bar z\\in Z$ with $\\pi(\\bar z)=\\bar x$ and changing to a subsequence of $\\{\\bar\\tau_k\\}\\subset T$ with $\\bar\\tau_k\\bar z$ convergent, this contradicts to Lemma \\ref{lem:sub}.\n\nNow suppose that $f_\\gamma(\\tau,x_\\gamma)$ is transient and $f_{\\gamma+1}(\\tau,x_{\\gamma+1})$ is recurrent.\nLet $x'\\in X_{\\gamma+1}$ be so that $(x',0)$ is $\\widetilde\\tau_{f_{\\gamma+1}}$-recurrent (cf. Remarks \\ref{rems:rec}).\nSince $(x',0)$ is cannot be $\\widetilde\\tau_{f_\\gamma\\circ\\pi_\\gamma^{\\gamma+1}}$-recurrent, there exist a neighbourhood $V\\subset X_{\\gamma+1}\\times\\mathbb R$ of $(x',0)$ and a replete semigroup $P\\subset T$ so that $\\widetilde\\tau_{f_\\gamma\\circ\\pi_\\gamma^{\\gamma+1}}(x',0)\\notin V$ for every $\\tau\\in P$.\nGiven an arbitrary compact set $C\\subset T$, by Theorem 6.32 in \\cite{G-H} there exists a replete semigroup $Q\\subset P\\setminus C$.\nSince $(x',0)$ is $\\widetilde\\tau_{f_{\\gamma+1}}$-recurrent, we can inductively construct a sequence $\\{\\tau_k\\}_{k\\geq 1}\\subset P$ with $\\tau_k x'\\to x'$, $f_{\\gamma+1}(\\tau_k,x')\\to0$, and $(f_\\gamma\\circ\\pi^{\\gamma+1}_\\gamma)(\\tau_k,x')\\nrightarrow 0$.\nThe point transitivity of $\\widetilde\\tau_{f_{\\gamma+1}}$ follows by the preceding statement.\n\\end{proof}\n\nFurthermore, we shall study the case of transfinite induction to a limit ordinal.\nThe arguments are quite similar, however with an approximation of a limit ordinal instead of an isometric group extension.\n\n\\begin{lemma}\\label{lem:lim}\nSuppose that $\\gamma$ is a limit ordinal with $1<\\gamma\\leq\\eta$.\n\\begin{enumerate}\n\\item If for every ordinal $1<\\alpha<\\gamma$ there exists an ordinal $\\alpha\\leq\\xi<\\gamma$ so that $f_\\xi(\\tau,x_\\xi)$ has a point transitive skew product extension, then $f_\\gamma(\\tau,x_\\gamma)$ has a point transitive skew product extension.\n\\item If there exist an ordinal $1\\leq\\alpha<\\gamma$ and a sequence $\\{(\\tau_k,x_k)\\}_{k\\geq 1}\\subset T\\times X_\\gamma$ with $d_\\gamma (x_k,\\tau_k x_k)\\to 0$  so that $(f_\\xi\\circ\\pi_\\xi^\\gamma)(\\tau_k,x_k)\\to 0$ for every $\\alpha\\leq\\xi<\\gamma$ and $f_\\gamma(\\tau_k,x_k)\\nrightarrow 0$ as $k\\to\\infty$ (or equivalently $(f_\\xi\\circ\\pi_\\xi^\\gamma)(\\tau_k,x_k)\\nrightarrow 0$ for every $\\alpha\\leq\\xi<\\gamma$ and $f_\\gamma(\\tau_k,x_k)\\to 0$), then $\\widetilde\\tau_{f_\\gamma}$ is necessarily point transitive.\n\\item If there exists an ordinal $1\\leq\\alpha<\\gamma$ so that for all $\\alpha\\leq\\xi<\\gamma$ the cocycle $f_\\xi(\\tau,x_\\xi)$ is transient, then $f_\\gamma(\\tau,x_\\gamma)$ is either transient or its skew product extension is point transitive.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nSuppose that the skew product of $\\tau_{f_\\gamma}$ on $X_\\gamma\\times\\mathbb R$ is not point transitive, and let $K\\subset\\mathbb R$ and $\\varepsilon>0$ be provided by Lemma \\ref{lem:at}.\nSince $\\gamma$ is a limit ordinal and $(X_\\gamma,T)$ is the inverse limit of the flows $\\{(X_\\xi,T):0\\leq\\xi<\\gamma\\}$, we can choose an ordinal $\\zeta<\\gamma$ so that for all $x,x'\\in X_\\gamma$ with $\\pi_\\zeta^\\gamma (x)=\\pi_\\zeta^\\gamma (x')$ holds $d_\\gamma (x,x')<\\varepsilon\/3$.\nIf we put $\\delta=\\varepsilon\/3$, then $d_\\gamma(x',\\tau x')<\\delta$ for $x'\\in X_\\gamma$ and $\\tau\\in T$ implies $d_\\gamma (x,\\tau x)<\\varepsilon$ for all $x\\in(\\pi^\\gamma_\\zeta)^{-1}(\\pi^\\gamma_\\zeta(x'))$.\nThese conditions remain valid even if the ordinal $\\zeta$ will be increased later.\nSince the $\\pi_\\zeta^{\\gamma}$-fibres are connected, for every $\\tau\\in T$ and $x'_\\gamma\\in X_\\gamma$ the range of $(f_{\\gamma}-f_\\zeta\\circ\\pi^{\\gamma}_\\zeta)(\\tau,x_\\gamma)$ on the $\\pi_\\zeta^{\\gamma}$-fibre of $x'_\\gamma$ is connected and contains $0$.\n\nUnder the hypothesis (i), we can choose $(\\tau,x_\\zeta)\\in T\\times X_\\zeta$ so that $f_\\zeta(\\tau,x_\\zeta)\\in 2K\\setminus K^0$ and $d_\\gamma(x'_\\gamma,\\tau x'_\\gamma)<\\delta$ for some $x_\\gamma'\\in(\\pi_\\zeta^{\\gamma})^{-1}(x_\\zeta)$.\nThus $d_\\gamma(x_\\gamma,\\tau x_\\gamma)<\\varepsilon$ holds for all $x_\\gamma\\in(\\pi_\\zeta^{\\gamma})^{-1}(x_\\zeta)$, and for $x_\\gamma$ with $(f_{\\gamma}-f_\\zeta\\circ\\pi^{\\gamma}_\\zeta)(\\tau,x_\\gamma)=0$ this contradicts $f_\\gamma(\\tau,x_\\gamma)\\notin 2K\\setminus K^0$.\nThus assertion (i) is verified.\n\nWe apply then Lemma \\ref{lem:sub} with $(Z,T)=(X_\\gamma,T)$, $(Y,T)=(X_\\zeta,T)$, $\\sigma=\\pi_\\zeta^{\\gamma}$, $h=(f_{\\gamma}-f_\\zeta\\circ\\pi^{\\gamma}_\\zeta)$, and $g=f_\\zeta$.\nHowever, given the sequence $\\{(\\tau_k,x_k)\\}_{k\\geq 1}\\subset T\\times X_\\gamma$ in hypothesis (ii), Lemma \\ref{lem:trans} provides a point $\\bar x\\in X_\\gamma$ and a sequence $\\{\\bar\\tau_k\\}\\subset T$ so that $(f_\\zeta\\circ\\pi_\\zeta^{\\gamma},f_{\\gamma}-f_\\zeta\\circ\\pi^{\\gamma}_\\zeta)(\\bar\\tau_k,\\bar x)=(g\\circ\\sigma,h)(\\bar\\tau_k,\\bar x)\\to(0,\\infty)$ (or $(\\infty,0)$) and $\\bar\\tau_k\\bar x\\to\\bar z$ as $k\\to\\infty$.\nThis is a contradiction to Lemma \\ref{lem:sub} and verifies (ii).\n\nNow suppose that $f_\\zeta(\\tau,x_\\zeta)$ is transient and $f_{\\gamma}(\\tau,x_{\\gamma})$ is recurrent, and choose $x'\\in X_\\gamma$ so that $(x',0)$ is $\\widetilde\\tau_{f_{\\gamma}}$-recurrent.\nSince $(x',0)$ is cannot be $\\widetilde\\tau_{f_\\zeta\\circ\\pi_\\zeta^{\\gamma}}$-recurrent, there exist a neighbourhood $V\\subset X_\\gamma\\times\\mathbb R$ of $(x',0)$ and a replete semigroup $P\\subset T$ with $\\widetilde\\tau_{f_\\zeta\\circ\\pi_\\zeta^{\\gamma}}(x',0)\\notin V$ for every $\\tau\\in P$.\nBy induction exists a sequence $\\{\\tau_k\\}_{k\\geq 1}\\subset P$ with $\\widetilde{(\\tau_k)}_{f_{\\gamma}}(x',0)\\to(x',0)$ as $k\\to\\infty$, and by Lemma \\ref{lem:trans} there exist a point $\\bar x\\in X_\\gamma$ and sequence $\\{\\bar\\tau_k\\}\\subset T$ so that $(f_\\gamma,f_\\zeta\\circ\\pi^{\\gamma}_\\zeta)(\\bar\\tau_k,\\bar x)=(g\\circ\\sigma+h,g\\circ\\sigma)(\\bar\\tau_k,\\bar x)\\to(0,\\infty)$ and $\\bar\\tau_k\\bar x\\to\\bar x$.\nThis contradiction to Lemma \\ref{lem:sub} verifies the statement (iii).\n\\end{proof}\n\n\\begin{proposition}\\label{prop:max}\nIf the real-valued cocycle $f(\\tau,x)$ is topologically recurrent apart from a coboundary, then there exists a maximal ordinal $1\\leq\\alpha\\leq\\eta$ so that the skew product extension $\\widetilde\\tau_{f_\\alpha}$ is point transitive on $X_\\alpha\\times\\mathbb R$.\nThe cocycle $(f-f_\\alpha\\circ\\pi_\\alpha) (\\tau,x)$ is relatively trivial with respect to $(f_\\alpha\\circ\\pi_\\alpha)(\\tau,x)$.\n\\end{proposition}\n\n\\begin{proof}\nLet us first suppose that the cocycle $f_\\xi(\\tau,x_\\xi)$ is recurrent for every ordinal $1\\leq\\xi<\\eta$, and let $\\mathcal O=\\{1\\leq\\xi\\leq\\eta:{f_\\xi}(\\tau,x_\\xi)\\enspace\\textup{is \\emph{not} a coboundary}\\}$.\nThis set is non-empty since $f_\\eta(\\tau,x)$ is not a coboundary, and let $\\beta$ be its minimal element.\nIf $\\beta=1$, then by Proposition \\ref{prop:isom} the recurrent skew product $\\widetilde{\\tau}_{f_1}$ of the isometric flow $(X_1,T)$ is point transitive.\nIf $\\beta>1$, then Fact \\ref{fact:GH} provides a sequence $\\{(\\tau_k,x_k)\\}_{k\\geq 1}\\subset T\\times X_\\beta$ with $d_\\beta(x_k,\\tau_k x_k)\\to 0$ and $f_\\beta(\\tau_k, x_k)\\to\\infty$.\nFor all $1\\leq\\zeta<\\beta$ holds $(f_\\beta\\circ\\pi_\\zeta^\\beta)(\\tau_k, x_k)\\to 0$, and by the Lemmas \\ref{lem:c_t} and \\ref{lem:lim} (ii) $\\widetilde{\\tau}_{f_\\beta}$ is point transitive.\n\nIf $f_\\xi(\\tau,x_\\xi)$ is transient for an ordinal $1\\leq\\xi<\\eta$, then let $\\beta$ be the minimal element of the set $\\mathcal O=\\{\\xi<\\zeta\\leq\\eta:f_\\zeta(\\tau,x_\\zeta)\\enspace\\textup{is topologically recurrent}\\}$.\nThis set is non-empty since $f_\\eta(\\tau,x_\\eta)$ is topologically recurrent, and it follows from the Lemmas \\ref{lem:c_t} and \\ref{lem:lim} (iii) that $\\widetilde\\tau_{f_\\beta}$ is even point transitive.\n\nNow let $\\mathcal O=\\{1\\leq\\xi\\leq\\eta:\\widetilde\\tau_{f_\\zeta}\\enspace\\textup{is \\emph{not} point transitive for all}\\enspace\\xi\\leq\\zeta\\leq\\eta\\}$.\nIf $\\mathcal O$ is empty, then $\\widetilde\\tau_{f_\\eta}$ is point transitive and $\\alpha=\\eta$.\nOtherwise, the set $\\mathcal O$ has a minimal element $\\gamma>1$ since $\\widetilde\\tau_{f_\\beta}$ is point transitive for some $1\\leq\\beta\\leq\\eta$.\nSince $\\gamma$ cannot be a limit ordinal by Lemma \\ref{lem:lim} (i), there exists a maximal ordinal $\\alpha\\geq 1$ with point transitive $\\widetilde\\tau_{f_\\alpha}$.\nIf $\\{(\\tau_k,x_k)\\}_{k\\geq 1}\\subset T\\times X$ is a sequence with $d (x_k,\\tau_k x_k)\\to 0$ and $(f_\\alpha\\circ\\pi_\\alpha)(\\tau_k,x_k)\\to 0$, then transfinite induction using the maximality of $\\alpha$ and Lemmas \\ref{lem:c_t}, \\ref{lem:lim} (ii) verifies that $(f_\\xi\\circ\\pi_\\xi)(\\tau_k,x_k)\\to 0$ for every $\\alpha\\leq\\xi\\leq\\eta$.\n\\end{proof}\n\nAfter the flow $(X_\\alpha,T)$ with a point transitive skew product extension $\\widetilde\\tau_{f_\\alpha}$ has been identified, we shall study the extension from $(X_\\alpha,T)$ to $(X,T)$.\nThere might be infinitely many isometric extensions in between, and therefore this extension is in general a distal extension.\nSince our construction will use the regulariser of this extension, it is necessary to leave the category of compact metric flows for the category of compact Hausdorff flows during the following construction (cf. Remark \\ref{rem:d_n_m}).\nHowever, the flow which will be constructed by means of the regulariser will be metric as a factor of the compact metric flow $(X,T)$.\n\n\\begin{proposition}\\label{prop:flow}\nThere exists a factor $(Y,T)=(X_\\alpha\\times M,T)=\\pi_Y(X,T)$ which is a Rokhlin extension of $(X_\\alpha,T)=\\rho_\\alpha(Y,T)$ by a distal minimal compact metric $\\mathbb R$-flow $(M,\\{\\phi^t:t\\in\\mathbb R\\})$ and the cocycle $f_\\alpha(\\tau,x_\\alpha)$ so that for every $x\\in X$ holds\n\\begin{equation}\\label{eq:p_X}\n\\pi_Y^{-1}(\\pi_Y(x))\\times\\{0\\}=\\mathcal D_{T, f_\\alpha\\circ\\pi_\\alpha}(x,0)\\cap(\\pi_\\alpha^{-1}(\\pi_\\alpha(x))\\times\\{0\\}) .\n\\end{equation}\nThe $\\mathbb R$-flow $\\{\\psi^t:t\\in\\mathbb R\\}\\subset\\textup{Aut}(Y,T)$ defined by $\\psi^t(x_\\alpha,m)=(x_\\alpha,\\phi^t(m))$ for $(x_\\alpha,m)\\in Y=X_\\alpha\\times M$ fulfils for every $y\\in Y$ and every $t\\in\\mathbb R$ that\n\\begin{equation}\\label{eq:o_flow}\n\\bar{\\mathcal O}_{T, f_\\alpha\\circ\\rho_\\alpha}(y,0)\\cap(\\rho_\\alpha^{-1}(\\rho_\\alpha(y))\\times\\{t\\})\\subset\\{(\\psi^t(y),t)\\} ,\n\\end{equation}\nwith coincidence of these sets if $(\\rho_\\alpha(y),0)\\in X_\\alpha\\times\\mathbb R$ is transitive for $\\widetilde\\tau_{f_\\alpha}$.\n\\end{proposition}\n\n\\begin{proof}\nWe shall construct a factor $(Y,T)$ of $(X,T)$ and a flow $\\{\\varphi^t:t\\in\\mathbb R\\}\\subset\\textup{Aut}(Y,T)$, and then we shall represent $(Y,T)$ as a Rokhlin extension of $(X_\\alpha,T)$.\nLet $(\\tilde X,T)$ be a distal minimal compact Hausdorff flow with $(X,T)=\\pi(\\tilde X,T)$ and a Hausdorff topological group $G\\subset\\textup{Aut}(\\tilde X,T)$ acting freely on the fibres of $\\pi_\\alpha\\circ\\pi$ so that $(X,T)$ is the $H$-orbit space of a subgroup $H\\subset G$ (cf. Fact \\ref{fact:reg_ex}).\nFor an arbitrary point $\\tilde z\\in \\tilde X$ and $t\\in\\mathbb R$ we define a closed subset of $G$ by\n\\begin{equation}\\label{eq:G}\nG_{\\tilde z,t}=\\{g\\in G: (\\pi(g(\\tilde z)),t)\\in\\mathcal D_{T,f_\\alpha\\circ\\pi_\\alpha}(\\pi(\\tilde z),0)\\} .\n\\end{equation}\nThe mapping $\\pi$ is open as a homomorphism of distal minimal compact flows, and hence for every $g\\in G_{\\tilde z,t}$ there exist nets $\\{\\tilde z_i\\}_{i\\in I}\\subset \\tilde X$ and $\\{\\tau_i\\}_{i\\in I}\\subset T$ with $\\tilde z_i\\to \\tilde z$, $\\tau_i\\pi(\\tilde z_i)\\to\\pi(g(\\tilde z))$, and $f_\\alpha(\\tau_i,\\pi_\\alpha\\circ\\pi(\\tilde z_i))\\to t$.\nSince the cocycle $(f_\\alpha\\circ\\pi_\\alpha)(\\tau,x_\\alpha)$ is constant on the fibres of $\\pi_\\alpha$ and $T$ is Abelian, it follows for every fixed $\\tau\\in T$ that\n\\begin{equation*}\n\\tau_i\\pi(\\tau \\tilde z_i)=\\tau_i\\tau\\pi(\\tilde z_i)=\\tau\\tau_i\\pi(\\tilde z_i)\\to\\tau\\pi(g(\\tilde z))=\\pi(\\tau g(\\tilde z))=\\pi(g(\\tau \\tilde z))\n\\end{equation*}\nand by the cocycle identity\n\\begin{eqnarray*}\nf_\\alpha(\\tau_i,\\pi_\\alpha\\circ\\pi(\\tau \\tilde z_i)) & = & f_\\alpha(\\tau_i,\\pi_\\alpha\\circ\\pi(\\tilde z_i))-f_\\alpha(\\tau,\\pi_\\alpha\\circ\\pi(\\tilde z_i))\\\\\n& & +f_\\alpha(\\tau,\\pi_\\alpha\\circ\\pi(\\tau_i\\tilde z_i))\\to t .\n\\end{eqnarray*}\nBy the density of the $T$-orbit of $\\tilde z$ and a diagonalisation of nets there exist for every $\\tilde x\\in \\tilde X$ nets $\\{\\tilde x_i\\}_{i\\in I}\\subset \\tilde X$ and $\\{\\tau'_i\\}_{i\\in I}\\subset T$ with $\\tilde x_i\\to \\tilde x$, $\\tau'_i\\pi(\\tilde x_i)\\to\\pi(g(\\tilde x))$, and $f_\\alpha(\\tau_i,\\pi_\\alpha\\circ\\pi(\\tilde x_i))\\to t$.\nTherefore\n\\begin{equation*}\n(\\pi(g(\\tilde x)),t)\\in\\mathcal D_{T,f_\\alpha\\circ\\pi_\\alpha}(\\pi(\\tilde x),0)\n\\end{equation*}\nso that $g\\in G_{\\tilde x,t}=G_{\\tilde z,t}=G_t$.\nBy symmetry follows now that $G_{-t}=(G_t)^{-1}$.\n\nThen we fix a point $x'\\in X$ with $\\bar{\\mathcal O}_{T,f_\\alpha}(\\pi_\\alpha(x'),0)=X_\\alpha\\times\\mathbb R$ and $\\mathcal D_{T,f_\\alpha\\circ\\pi_\\alpha}(x',0)=\\bar{\\mathcal O}_{T,f_\\alpha\\circ\\pi_\\alpha}(x',0)$ (cf. Fact \\ref{fact:o_p}).\nThe set $G_t$ is non-empty for every $t\\in\\mathbb R$, since $\\bar{\\mathcal O}_{T, f_\\alpha}(\\pi_\\alpha(x'),0)=X_\\alpha\\times\\mathbb R$ and the compactness of $X$ ensure that\n\\begin{equation*}\n\\bar{\\mathcal O}_{T,f_\\alpha\\circ\\pi_\\alpha}(x',0)\\cap\\pi_\\alpha^{-1}(\\pi_\\alpha(x'))\\times\\{t\\}\\neq\\emptyset .\n\\end{equation*}\nFor arbitrary $t,t'\\in\\mathbb R$ and $g\\in G_t$, $g'\\in G_{t'}$, we select $\\tilde x, \\tilde z\\in \\tilde X$ so that $\\pi(\\tilde x)=x'$ and $\\tilde x=g'(\\tilde z)$.\nThen we have\n\\begin{equation*}\n(x',t')=(\\pi(g'(\\tilde z)),t')\\in\\mathcal D_{T,f_\\alpha\\circ\\pi_\\alpha}(\\pi(\\tilde z),0) ,\n\\end{equation*}\nand for $\\tilde y=g(\\tilde x)=gg'(\\tilde z)$ it holds that $(\\pi(\\tilde y),t)\\in\\bar{\\mathcal O}_{T,f_\\alpha\\circ\\pi_\\alpha}(x',0)=\\mathcal D_{T,f_\\alpha\\circ\\pi_\\alpha}(x',0)$.\nBy Remark \\ref{rem:o_p} follows $(\\pi(\\tilde y),t+t')\\in\\mathcal D_{T,f_\\alpha\\circ\\pi_\\alpha}(\\pi(\\tilde z),0)$ so that $gg'\\in G_{t+t'}$.\nHence $G_t G_{t'}\\subset G_{t+t'}$ holds for all $t,t'\\in\\mathbb R$, and from $G_{-t}=(G_{t})^{-1}$ follows $(G_{t})^{-1} G_{t+t'}=G_{-t} G_{t+t'}\\subset G_{t'}$ so that $G_{t} G_{t'}= G_{t+t'}$.\nThus the Hausdorff topological group\n\\begin{equation*}\n\\tilde G=\\cup_{t\\in\\mathbb R}G_t\n\\end{equation*}\nhas the closed set $G_0\\supset H$ as a normal subgroup so that $G_t$ is a $G_0$-coset in $\\tilde G$ for every $t\\in\\mathbb R$.\nMoreover, the mapping $t\\mapsto G_t$ is a group homomorphism from $\\mathbb R$ into $\\tilde G\/G_0$.\nThe group $G_0$ is not necessarily compact, however its orbit space on $\\tilde X$ defines a partition into sets invariant under $H\\subset G_0$.\nHence this is also a partition of $X$, and the equivalence relation $R_Y$ of this partition of $X$ is $T$-invariant since $G_0\\subset\\textup{Aut}(\\tilde X,T)$.\nMoreover, $R_Y$ is closed in $X^2$, since definition (\\ref{eq:G}) implies that $(x,x')\\in R_Y$ if and only if $(x',0)\\in\\mathcal D_{T, f_\\alpha\\circ\\pi_\\alpha}(x,0)\\cap(\\pi_\\alpha^{-1}(\\pi_\\alpha(x))\\times\\{0\\})$.\nThe factor $(Y,T)=\\pi_Y(X,T)$ defined by the $T$-invariant closed equivalence relation $R_Y$ is an extension of $(X_\\alpha,T)=\\rho_\\alpha(Y,T)$, and equality (\\ref{eq:p_X}) follows.\nThe $\\mathbb R$-action $\\{\\varphi^t:t\\in\\mathbb R\\}\\subset\\textup{Aut}(Y,T)$ is well defined for every $y\\in Y$ and $t\\in\\mathbb R$ by\n\\begin{equation*}\n\\varphi^t(y)=G_t ((\\pi_Y\\circ\\pi)^{-1}(y))=G_t (\\{\\tilde x\\in \\tilde X:G_0(\\tilde x)=y\\}) .\n\\end{equation*}\nLet $\\{(t_k,y_k)\\}_{k\\geq 1}\\subset\\mathbb R\\times Y$ be a sequence with $(t_k,y_k)\\to (t,y)$, then $\\varphi^{t_k}(y_k)=G_0 g_k(\\tilde x_k)$ for a sequence $\\{\\tilde x_k\\}_{k\\geq 1}\\subset\\tilde X$ with $\\pi_Y\\circ\\pi(\\tilde x_k)=y_k$ and $g_k\\in G_{t_k}$.\nWe can assume that $\\tilde x_k\\to\\tilde x$ and $g_k(\\tilde x_k)\\to\\tilde z$ so that $(\\pi(\\tilde z),t)\\in\\mathcal D_{T,f_\\alpha\\circ\\pi_\\alpha}(\\pi(\\tilde x),0)$ and $\\tilde z=g_t(\\tilde x)$ for some $g_t\\in G_t$.\nFrom $\\pi_Y\\circ\\pi(\\tilde x)=y$ and $\\varphi^{t_k}(y_k)=\\pi_Y\\circ\\pi(g_k(\\tilde x_k))\\to\\pi_Y\\circ\\pi(\\tilde z)=\\varphi^t(y)$ follows the continuity of the action $\\{\\varphi^t:t\\in\\mathbb R\\}$ on $Y$.\n\nWe turn to the inclusion (\\ref{eq:o_flow}).\nSuppose that $(y_i,t)\\in\\bar{\\mathcal O}_{T, f_\\alpha\\circ\\rho_\\alpha}(y,0)\\cap\\rho_\\alpha^{-1}(x_\\alpha)\\times\\{t\\}$ for some $x_\\alpha\\in X_\\alpha$ and $i\\in\\{1,2\\}$, and select $x\\in\\pi_Y^{-1}(y)$.\nBy the compactness of $X$ there exist points $x_i\\in\\pi_Y^{-1}(y_i)\\subset\\pi_\\alpha^{-1}(x_\\alpha)$ so that $(x_i,t)\\in\\bar{\\mathcal O}_{T, f_\\alpha\\circ\\pi_\\alpha}(x,0)$, and therefore $(x_2,0)\\in\\mathcal D_{T,f_\\alpha\\circ\\pi_\\alpha}(x_1,0)$.\nThe equality (\\ref{eq:p_X}) implies that $y_1=\\pi_Y(x_1)=\\pi_Y(x_2)=y_2$, and thus for every $y\\in Y$ and $t\\in\\mathbb R$ holds\n\\begin{equation}\\label{eq:card_1}\n\\textup{card}\\{\\bar{\\mathcal O}_{T, f_\\alpha\\circ\\rho_\\alpha}(y,0)\\cap\\rho_\\alpha^{-1}(\\rho_\\alpha(y))\\times\\{t\\}\\}\\leq 1 .\n\\end{equation}\nMoreover, for $x_\\alpha=\\rho_\\alpha(y)$ follows $x_1=\\pi(g_t(\\tilde x))$ with $g_t\\in G_t$ and $\\tilde x\\in\\pi^{-1}(x)\\subset\\tilde X$.\nHence $y_1=\\pi_Y(x_1)=\\varphi^t(y)$ and inclusion (\\ref{eq:o_flow}) is verified.\nFor $\\widetilde\\tau_{f_\\alpha}$-transitive $(\\rho_\\alpha(y),0)$ the cardinality in (\\ref{eq:card_1}) is equal to $1$ for every $y\\in Y$ and $t\\in\\mathbb R$, and for $y'\\in\\rho_\\alpha^{-1}(\\rho_\\alpha(y))$ and $x_\\alpha\\in X_\\alpha$ holds $\\bar{\\mathcal O}_{T,f_\\alpha\\circ\\rho_\\alpha}(y',0)\\cap\\rho_\\alpha^{-1}(x_\\alpha)\\times\\{0\\}=\\{(y_1,0)\\}$.\nWe fix a point $\\bar x\\in X_\\alpha$ with $\\widetilde\\tau_{f_\\alpha}$-transitive $(\\bar x,0)$.\nIf $(y_2,0)\\in\\mathcal D_{T,f_\\alpha\\circ\\rho_\\alpha}(y',0)\\cap\\rho_\\alpha^{-1}(x_\\alpha)\\times\\{0\\}$, then Remark \\ref{rem:o_p} implies that $(y_2,0)\\in\\mathcal D_{T,f_\\alpha\\circ\\rho_\\alpha}(y_1,0)$, and as above follows $y_1=y_2$.\nHence\n\\begin{equation}\\label{eq:homeo_Y}\n\\bar{\\mathcal O}_{T,f_\\alpha\\circ\\rho_\\alpha}(y',0)\\cap\\rho_\\alpha^{-1}(x_\\alpha)\\times\\{0\\}=\\mathcal D_{T,f_\\alpha\\circ\\rho_\\alpha}(y',0)\\cap\\rho_\\alpha^{-1}(x_\\alpha)\\times\\{0\\}\n\\end{equation}\nholds for every $y'\\in\\rho_\\alpha^{-1}(\\bar x)$ and $x_\\alpha\\in X_\\alpha$.\nFor distinct $y',y''\\in\\rho_\\alpha^{-1}(\\bar x)$ follows\n\\begin{equation*}\n\\bar{\\mathcal O}_{T,f_\\alpha\\circ\\rho_\\alpha}(y',0)\\cap\\bar{\\mathcal O}_{T,f_\\alpha\\circ\\rho_\\alpha}(y'',0)\\cap\\rho_\\alpha^{-1}(x_\\alpha)\\times\\{0\\}=\\emptyset .\n\\end{equation*}\nIndeed, given a point $\\bar y$ in this intersection, for every sequence $\\{\\tau_k\\}_{k\\geq 1}\\subset T$ with $\\tau_k\\bar x\\to\\rho_\\alpha(\\bar y)$ and $f_\\alpha(\\tau_k,\\bar x)\\to 0$ follows by equality (\\ref{eq:card_1}) that $d_Y(\\tau_k y',\\tau_k y'')\\to 0$, in contradiction to the distality of $(Y,T)$.\nHence the mapping $\\iota:X_\\alpha\\times\\rho_\\alpha^{-1}(\\bar x)\\longrightarrow Y$\n\\begin{equation*}\n(x_\\alpha,y')\\mapsto\\rho_\\alpha^{-1}(x_\\alpha)\\cap\\{y\\in Y:(y,0)\\in\\bar{\\mathcal O}_{T,f_\\alpha\\circ\\rho_\\alpha}(y',0)\\}\n\\end{equation*}\nis well-defined, one-to-one, and by equality (\\ref{eq:homeo_Y}) also continuous.\nFor a dense set of points $\\bar y\\in Y$ holds the $\\widetilde\\tau_{f_\\alpha}$-transitivity of $(\\rho_\\alpha(\\bar y),0)$, since $\\rho_\\alpha$ is open.\nWe can conclude for every $y\\in Y$ that $\\mathcal D_{T,f_\\alpha\\circ\\rho_\\alpha}(y,0)\\cap\\rho_\\alpha^{-1}(\\bar x)\\times\\{0\\}\\neq\\emptyset$, and thus $\\mathcal D_{T,f_\\alpha\\circ\\rho_\\alpha}(y',0)\\cap\\{y\\}\\times\\{0\\}=\\bar{\\mathcal O}_{T,f_\\alpha\\circ\\rho_\\alpha}(y',0)\\cap\\{y\\}\\times\\{0\\}\\neq\\emptyset$ for some $y'\\in\\rho_\\alpha^{-1}(\\bar x)$.\nHence $\\iota$ is onto and by compactness $Y$ and $X_\\alpha\\times\\rho_\\alpha^{-1}(\\bar x)$ are homeomorphic.\n\nLet $\\{\\phi^t:t\\in\\mathbb R\\}$ be the restriction of $\\{\\varphi^t:t\\in\\mathbb R\\}$ to the $\\{\\varphi^t:t\\in\\mathbb R\\}$-invariant compact metric space $M=\\rho_\\alpha^{-1}(\\bar x)$.\nFor every $y'\\in M$ and $\\tau\\in T$ holds $\\bar{\\mathcal O}_{T, f_\\alpha\\circ\\rho_\\alpha}(y',0)\\cap\\rho_\\alpha^{-1}(\\bar x)\\times\\{-f_\\alpha(\\tau,\\bar x) \\}=\\{(\\phi^{-f_\\alpha(\\tau,\\bar x)}(y'),-f_\\alpha(\\tau,\\bar x))\\}$ and $\\widetilde{\\tau}_{f_\\alpha\\circ\\pi_Y}(\\phi^{-f_\\alpha(\\tau,\\bar x)}(y'),-f_\\alpha(\\tau,\\bar x))\\in\\rho_\\alpha^{-1}(\\tau\\bar x)\\cap\\{y\\in Y:(y,0))\\in\\bar{\\mathcal O}_{T,f_\\alpha\\circ\\rho_\\alpha}(y',0)\\}$.\nTherefore $\\tau\\phi^{-f_\\alpha(\\tau,\\bar x)}(y')=\\iota(\\tau\\bar x,y')$ and $\\tau y=\\iota(\\tau_{\\phi,f_\\alpha}(\\bar x,y))$ for every $y\\in M$ and $\\tau\\in T$.\nThe minimality of $(Y,T)$ implies that $(X_\\alpha\\times M,T)$ and $(Y,T)$ are topologically isomorphic via $\\iota$.\nMoreover, for the mapping $\\psi^t(x_\\alpha,m)=(x_\\alpha,\\phi^t(m))$ with $\\psi\\in\\textup{Aut}(X_\\alpha\\times M,T)$ and every $m\\in M=\\rho_\\alpha^{-1}(\\bar x)$ and $t\\in\\mathbb R$ holds $\\varphi^t(\\iota(\\tau_{\\phi,f_\\alpha}(\\bar x,m)))=\\varphi^t(\\tau m)=\\tau\\varphi^t(m)=\\iota(\\tau_{\\phi,f_\\alpha}(\\bar x,\\phi^t(m)))=\\iota(\\psi^t(\\tau_{\\phi,f_\\alpha}(\\bar x,m)))$.\nBy the minimality of $(X_\\alpha\\times M,T)$ follows $\\psi^t=\\iota^{-1}\\circ\\varphi^t\\circ\\iota$ for every $t\\in\\mathbb R$.\nThe flow $(M,\\{\\phi^t:t\\in\\mathbb R\\})$ is minimal and distal, since a non-transitive point $m'\\in M$ and a proximal pair $(m',m'')\\in M^2$, respectively, give rise a non-transitive point $(x_\\alpha,m')\\in Y$ and a proximal pair $((x_\\alpha,m'),(x_\\alpha,m''))\\in Y^2$, respectively.\n\\end{proof}\n\nIt should be mentioned that an ordinal $\\xi\\leq\\eta$ with $(Y,T)=(X_\\xi,T)$ does not necessarily exist.\nTherefore we shall define a cocycle $f_Y:T\\times Y\\longrightarrow\\mathbb R$ independently of the cocycles $f_\\xi(\\tau,x_\\xi)$, and it will turn out that $(f_Y\\circ\\pi_Y)(\\tau,x)$ can be chosen topologically cohomologous to $f$.\n\n\\begin{proposition}\\label{prop:res}\nThere exists a topological cocycle $f_Y(\\tau,y)$ of the flow $(Y,T)$ so that $(f_Y\\circ\\pi_Y)(\\tau,x)$ is topologically cohomologous to $f(\\tau,x)$ and $f_Y(\\tau,y)$ is relatively trivial with respect to $(f_\\alpha\\circ\\rho_\\alpha)(\\tau,y)$.\n\\end{proposition}\n\nWe shall prove another technical lemma first.\n\n\\begin{lemma}\\label{lem:tilde_coc}\nLet $(Z,T)$ be a distal minimal compact metric flow which extends $(X_\\alpha,T)=\\sigma_\\alpha(Z,T)$, and let $G\\subset\\textup{Aut}(Z,T)$ be a Hausdorff topological group preserving the fibres of $\\sigma_\\alpha$.\nSuppose that there exists a continuous group homomorphism $\\varphi:G\\longrightarrow\\mathbb R$ so that for every $g\\in G$ and every $z\\in Z$ it holds that\n\\begin{equation*}\n(g(z),\\varphi(g))\\in\\mathcal D_{T,f_\\alpha\\circ\\sigma_\\alpha}(z,0) .\n\\end{equation*}\nFurthermore, suppose that $h(\\tau,z)$ is a real-valued cocycle of $(Z,T)$ which is relatively trivial with respect to $(f_\\alpha\\circ\\sigma_\\alpha)(\\tau,z)$.\nThen there exists a continuous cocycle $\\bar h((\\tau,g),z)$ of the flow $(Z,T\\times G)$ with the action $\\{g\\circ\\tau:(\\tau,g)\\in T\\times G\\}$ so that $h(\\tau,z)=\\bar h((\\tau,\\mathbf 1_G),z)$ holds for every $(\\tau,z)\\in T\\times Z$ and the mapping $h\\mapsto\\bar h$ is linear.\nFor $z\\in Z$, $g\\in G$, and a sequence $\\{(\\tau_k,z_k)\\}_{k\\geq 1}\\subset T\\times Z$ with $z_k\\to z$, $\\tau_k z_k\\to g(z)$, and $(f_\\alpha\\circ\\rho_\\alpha)(\\tau_k,z_k)\\to\\varphi(g)$ holds\n\\begin{equation}\\label{eq:h_uni}\nh(\\tau_k,z_k)\\to\\bar h((\\mathbf 1_T,g),z)\\quad\\textup{as}\\enspace k\\to\\infty.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nWe put $F=(f_\\alpha\\circ\\sigma_\\alpha,h):T\\times Z\\longrightarrow\\mathbb R^2$ and fix a point $\\bar z\\in Z$ so that $\\bar{\\mathcal O}_{T,F} (\\bar z,0,0)$ and $\\mathcal D_{T,F} (\\bar z,0,0)$ coincide in $Z\\times(\\mathbb R_\\infty)^2$ (cf. Fact \\ref{fact:o_p}).\nFor every $g\\in G$ we fix a sequence $\\{\\tau_k^g\\}_{k\\geq 1}\\subset T$ with $\\widetilde{(\\tau_k^g)}_{f_\\alpha\\circ\\sigma_\\alpha}(\\bar z,0)\\to(g(\\bar z),\\varphi(g))$ as $k\\to\\infty$, with $\\{\\tau_k^{\\mathbf 1_G}=\\mathbf 1_T\\}_{k\\geq 1}$.\nSince $g\\in\\textup{Aut}(Z,T)$ and $f_\\alpha\\circ\\sigma_\\alpha\\circ g=f_\\alpha\\circ\\sigma_\\alpha$, we can conclude for every $t\\in T$ that $\\tau_k^gt\\bar z=t\\tau_k^g\\bar z\\to t g(\\bar z)=g(t\\bar z)$ as $k\\to\\infty$ as well as\n\\begin{equation}\\label{eq:transf}\n(f_\\alpha\\circ\\sigma_\\alpha)(\\tau_k^g,t\\bar z)=(f_\\alpha\\circ\\sigma_\\alpha)(t, \\tau_k^g\\bar z)+(f_\\alpha\\circ\\sigma_\\alpha)(\\tau_k^g,\\bar z)-(f_\\alpha\\circ\\sigma_\\alpha)(t,\\bar z)\\to\\varphi(g).\n\\end{equation}\nBy the relative triviality of $h(\\tau,z)$ with respect to $(f_\\alpha\\circ\\sigma_\\alpha)(\\tau,z)$, the sequence $\\{h(\\tau\\tau_k^g,t\\bar z)\\}_{k\\geq 1}$ converges for all $\\tau,t\\in T$.\nThus we can put\n\\begin{equation}\\label{eq:def_g}\n\\bar h((\\tau,g),t\\bar z)=\\lim_{k\\to\\infty} h(\\tau\\tau_k^g,t\\bar z)=h(\\tau,g(t\\bar z))+\\lim_{k\\to\\infty} h(\\tau_k^g,t\\bar z)\n\\end{equation}\nfor every $(\\tau, g,t\\bar z)\\in T\\times G\\times Z$.\nSuppose that there exist sequences $\\{(\\tau_k^{i},z_k^i)\\}_{k\\geq 1}\\subset T\\times Z$ for $i=1,2$ so that $z_k^i\\to z$, $\\tau_k^i z_k^i\\to g(z)=\\lim_{k\\to\\infty}g(z_k^i)$, and\n\\begin{equation*}\n(f_\\alpha\\circ\\sigma_\\alpha)(\\tau_k^{i},z_k^i)\\to\\varphi(g) \\quad\\textup{as}\\enspace k\\to\\infty,\n\\end{equation*}\nwhile for $i=1,2$ the limit points $\\bar h_i=\\lim_{k\\to\\infty} h(\\tau_k^{i},z_k^i)\\in\\mathbb R_\\infty$ are either distinct or both equal to $\\infty$.\nThen $(g(z),\\varphi(g),\\bar h_i)\\in\\mathcal D_{T,F} (z,0,0)$ for $i=1,2$, and for every $\\tau'\\in T$ follows from $g\\in\\textup{Aut}(Z,T)$ and the cocycle identity that\n\\begin{eqnarray*}\n(g(\\tau' z),\\varphi(g)+(f_\\alpha\\circ\\sigma_\\alpha)(\\tau',g(z))-(f_\\alpha\\circ\\sigma_\\alpha)(\\tau',z),h(\\tau',g(z))+\\bar h_i-h(\\tau',z))=\\hspace{-5mm}\\\\\n(g(\\tau' z),\\varphi(g),h(\\tau',g(z))+\\bar h_i-h(\\tau',z))\\in\\mathcal D_{T,F}(\\tau' z,0,0).\n\\end{eqnarray*}\nSince $\\bar{\\mathcal O}_T(z)=Z$, either there are distinct points $a_1,a_2\\in\\mathbb R_\\infty$ with $(g(\\bar z),\\varphi(g),a_i)\\in\\mathcal D_{T,F} (\\bar z,0,0)$ or it holds that $(g(\\bar z),\\varphi(g),\\infty)\\in\\mathcal D_{T,F} (\\bar z,0,0)$.\nIn either case, since $\\bar{\\mathcal O}_{T,F} (\\bar z,0,0)=\\mathcal D_{T,F} (\\bar z,0,0)$ in $Z\\times(\\mathbb R_\\infty)^2$, this contradicts to the relative triviality of $h(\\tau,z)$ with respect to $(f_\\alpha\\circ\\sigma_\\alpha)(\\tau,z)$.\nTherefore equality (\\ref{eq:h_uni}) holds true, and the definition (\\ref{eq:def_g}) extends uniquely from the $T$-orbit of $\\bar z$ to a continuous mapping $\\bar h:T\\times G\\times Z\\longrightarrow\\mathbb R$ since the action of $T\\times G$ on $X$ and $\\varphi$ are continuous.\n\nFor the cocycle identity let $(\\tau_1,g_1),(\\tau_2,g_2)\\in T\\times G$ be arbitrary with sequences $\\{\\tau_k^{g_1}\\}_{k\\geq 1},\\{\\tau_k^{g_2}\\}_{k\\geq 1}\\subset T$.\nBy equality (\\ref{eq:transf}) we select a sequence $\\{k_l\\}_{l\\geq 1}\\subset\\mathbb N$ with\n\\begin{equation*}\n\\tau_{k_l}^{g_2}\\tau_l^{g_1}\\bar z=\\tau_l^{g_1}\\tau_{k_l}^{g_2}\\bar z\\to g_2(g_1(\\bar z))\\enspace\\textup{and}\\enspace (f_\\alpha\\circ\\sigma_\\alpha)(\\tau_l^{g_1}\\tau_{k_l}^{g_2},\\bar z)\\to\\varphi(g_2)+\\varphi(g_1)=\\varphi(g_2 g_1)\n\\end{equation*}\nas $l\\to\\infty$.\nThus we can put $\\{\\tau_l^{g_2 g_1}\\}_{l\\geq 1}=\\{\\tau_l^{g_1}\\tau_{k_l}^{g_2}\\}_{l\\geq 1}$, and for every $t\\in T$ the equality (\\ref{eq:transf}) implies that $\\tau_{k_l}^{g_2}\\tau_2 t\\bar z\\to g_2(\\tau_2 t\\bar z)$, $\\tau_l^{g_1}\\tau_{k_l}^{g_2}\\tau_2 t\\bar z\\to(g_2 g_1)(\\tau_2 t\\bar z)$, and\n\\begin{eqnarray*}\n(f_\\alpha\\circ\\sigma_\\alpha)(\\tau_1\\tau_l^{g_1},\\tau_{k_l}^{g_2}\\tau_2 t\\bar z)=\\hspace{7.5cm}\\\\\n(f_\\alpha\\circ\\sigma_\\alpha)(\\tau_1,\\tau_l^{g_1}\\tau_{k_l}^{g_2}\\tau_2 t\\bar z)+(f_\\alpha\\circ\\sigma_\\alpha)(\\tau_l^{g_1}\\tau_{k_l}^{g_2},\\tau_2 t\\bar z)-(f_\\alpha\\circ\\sigma_\\alpha)(\\tau_{k_l}^{g_2},\\tau_2 t\\bar z)\\\\\n\\to (f_\\alpha\\circ\\sigma_\\alpha)(\\tau_1,g_2(\\tau_2 t\\bar z))+\\varphi(g_2g_1)-\\varphi(g_2) \\quad\\textup{as}\\enspace l\\to\\infty.\n\\end{eqnarray*}\nThe uniqueness according to equality (\\ref{eq:h_uni}) verifies that $\\bar h((\\tau_1,g_2 g_1 g_2^{-1}),g_2(\\tau_2 t\\bar z))=\\lim_{l\\to\\infty}h(\\tau_1\\tau_l^{g_1},\\tau_2\\tau_{k_l}^{g_2}t\\bar z)$, and therefore\n\\begin{eqnarray*}\n\\bar h((\\tau_1,g_2 g_1 g_2^{-1}),g_2(\\tau_2t\\bar z))+\\bar h((\\tau_2,g_2),t\\bar z)=\\hspace{4.5cm}\\\\\n=\\lim_{l\\to\\infty}h(\\tau_1\\tau_l^{g_1},\\tau_2\\tau_{k_l}^{g_2}t\\bar z)+\\lim_{l\\to\\infty}h(\\tau_2\\tau_{k_l}^{g_2},t\\bar z)=\\\\\n=\\lim_{l\\to\\infty}h(\\tau_1\\tau_l^{g_1}\\tau_2\\tau_{k_l}^{g_2},t\\bar z)=\\bar h((\\tau_1\\tau_2,g_2g_1),t\\bar z) .\n\\end{eqnarray*}\nWe substitute $g_2^{-1} g_1 g_2$ for $g_1$ and obtain from $\\bar{\\mathcal O}_T(\\bar z)=Z$ and the continuity of $\\bar h$ that cocycle identity is valid.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{prop:res}]\nLet $(Y_c,T)=\\pi_c(X,T)$ be the flow defined by the connected components of the fibres of $\\pi_Y$ (cf. \\cite{MMWu}, Definition 2.2), and let $\\rho$ be the homomorphism from $(Y_c,T)$ onto $(Y,T)=\\rho(Y_c,T)$.\nWith a \\emph{RIM} $\\{\\mu_{c,y}:y\\in Y_c\\}$ for the distal extension $(Y_c,T)=\\pi_c(X,T)$ we define a cocycle $f_c(\\tau,y)=\\mu_{c,y}(f(\\tau,\\cdot))$ for every $(\\tau,y)\\in T\\times Y_c$.\nWe fix a point $\\bar x\\in X$ with $\\mathcal D_{T,f_\\alpha\\circ\\pi_\\alpha}(\\tau\\bar x,0)=\\bar{\\mathcal O}_{T,f_\\alpha\\circ\\pi_\\alpha}(\\tau\\bar x,0)$ for all $\\tau\\in T$.\nBy equality (\\ref{eq:p_X}) and $\\pi_c^{-1}(\\pi_c(\\tau\\bar x))\\subset\\pi_Y^{-1}(\\pi_Y(\\tau\\bar x))$ holds for all $\\tau\\in T$\n\\begin{equation*}\n\\bar{\\mathcal O}_{T,f_\\alpha\\circ\\pi_\\alpha}(\\bar x,0)\\cap(\\pi_c^{-1}(\\pi_c(\\tau\\bar x))\\times\\mathbb R)=\\pi_c^{-1}(\\pi_c(\\tau\\bar x))\\times\\{(f_\\alpha\\circ\\pi_\\alpha)(\\tau,\\bar x)\\} .\n\\end{equation*}\nLet $F(\\tau,x)$ be the $\\mathbb R^2$-valued cocycle $(f_\\alpha\\circ\\pi_\\alpha,f)$.\nWe shall verify that\n\\begin{eqnarray}\\label{eq:f_c}\n\\bar{\\mathcal O}_{T,F} (\\bar x,0,0)\\cap(\\pi_c^{-1}(\\pi_c(\\tau\\bar x))\\times\\{(f_\\alpha\\circ\\pi_\\alpha)(\\tau,\\bar x)\\}\\times\\mathbb R)=\\nonumber\\\\\n\\{(x,(f_\\alpha\\circ\\pi_\\alpha)(\\tau,\\bar x),b_\\tau(x)):x\\in\\pi_c^{-1}(\\pi_c(\\tau\\bar x))\\}\n\\end{eqnarray}\nfor every $\\tau\\in T$, in which $b_\\tau:\\pi_c^{-1}(\\pi_c(\\tau\\bar x))\\longrightarrow\\mathbb R$ is a continuous function.\nIndeed, for a sequence $\\{\\tau_k\\}_{k\\geq 1}\\subset T$ with $\\tau_k\\tau\\bar x\\to x\\in\\pi_c^{-1}(\\pi_c(\\tau\\bar x))$ and $(f_\\alpha\\circ\\pi_\\alpha)(\\tau_k,\\tau\\bar x)\\to 0$ follows by the relative triviality of $(f-f_\\alpha\\circ\\pi_\\alpha)(\\tau,x)$ the existence and uniqueness of the limit $b_\\tau(x)$ of $f(\\tau_k,\\tau\\bar x)$.\nIt also follows that for every $\\varepsilon>0$ there exists a $\\delta>0$ so that for all $\\tau\\in T$ and $x,x'\\in\\pi_c^{-1}(\\pi_c(\\tau\\bar x))$ with $d(x,x')<\\delta$ holds $|b_\\tau(x)-b_\\tau(x')|<\\varepsilon$.\nSince the fibres of $\\pi_c$ are connected, a covering of $X$ by $\\delta$-neighbourhoods provides a constant $D>0$ with $|b_\\tau(x)-b_\\tau(x')|<D$ for all $\\tau\\in T$ and $x,x'\\in\\pi_c^{-1}(\\pi_c(\\tau\\bar x))$.\nEquality (\\ref{eq:f_c}) shows for $x\\in\\pi_c^{-1}(\\pi_c(\\bar x))$ and $\\tau\\in T$ that $b_{\\mathbf 1_T}(x)+f(\\tau,x)=b_{\\tau}(\\tau x)$ and hence $|f(\\tau,\\bar x)-f(\\tau,x)|\\leq 2D$.\nSince $(f-f_c\\circ\\pi_c)(\\tau,x)$ assumes zero on each $\\pi_c$-fibre, for all $\\tau\\in T$ holds $|(f-f_c\\circ\\pi_c)(\\tau,\\bar x)|< 2D$ so that this is a coboundary.\n\nDue to Theorem 3.7 in \\cite{MMWu} the extension from $(Y,T)$ to $(Y_c,T)$ is an isometric extension, and by Fact \\ref{fact:reg_ex} there exists a compact group extension $(\\tilde Y,T)$ of $(Y,T)=\\rho(Y_c,T)$ by $G\\subset\\textup{Aut}(\\tilde Y,T)$ so that $(Y_c,T)=\\sigma(\\tilde Y,T)$ is the orbit space of a compact subgroup $H\\subset G$.\nFor every sequence $\\{(\\tau_k,\\tilde y_k)\\}_{k\\geq 1}\\subset T\\times\\tilde Y$ with $d_{\\tilde Y} (\\tilde y_k,\\tau_k\\tilde y_k)\\to 0$ and $(f_\\alpha\\circ\\rho_\\alpha\\circ\\rho\\circ\\sigma)(\\tau_k,\\tilde y_k)\\to 0$ holds $(f_c\\circ\\sigma)(\\tau_k,\\tilde y_k)\\to 0$.\nOtherwise, by Lemma \\ref{lem:trans} there exists a sequence $\\{(\\tilde\\tau_k,x_k)\\}_{k\\geq 1}\\subset T\\times X$ so that $d(x_k,\\tilde\\tau_k x_k)\\to 0$, $(f_\\alpha\\circ\\pi_\\alpha)(\\tilde\\tau_k,x_k)\\to 0$, and $(f_c\\circ\\pi_c)(\\tilde\\tau_k,x_k)\\to\\infty$, which contradicts to Proposition \\ref{prop:max} and the boundedness of the transfer function between $f$ and $f_c\\circ\\pi_c$.\nWe can apply Lemma \\ref{lem:tilde_coc} for the flow $(\\tilde Y,T)$, the cocycle $h=f_c\\circ\\sigma$, the group $G\\subset\\textup{Aut}(\\tilde Y,T)$, and the group homomorphism $\\varphi\\equiv 0$, and we obtain a real valued cocycle $\\bar h((\\tau,h),\\tilde y)$ with $\\bar h((\\tau,\\mathbf 1_G),\\tilde y)=(f_c\\circ\\sigma)(\\tau,\\tilde y)$ for every $(\\tau,\\tilde y)\\in T\\times\\tilde Y$.\nWe define a topological cocycle of $(Y,T)$ by $f_Y(\\tau,y)=\\mu_{Y,y}((f_c\\circ\\sigma)(\\tau,\\cdot))$, where $\\{\\mu_{Y,y}:y\\in Y\\}$ is the \\emph{RIM} for the extension $(Y,T)=\\rho\\circ\\sigma(\\tilde Y,T)$.\nFrom the cocycle identity $\\bar h((\\tau,g),\\tilde y)=(f_c\\circ\\sigma)(\\tau,g(\\tilde y))+\\bar h((\\mathbf 1_T,g),\\tilde y)=\\bar h((\\mathbf 1_T,g),\\tau\\tilde y)+(f_c\\circ\\sigma)(\\tau,\\tilde y)$ and the boundedness of $\\bar h((\\mathbf 1_T,g),\\tilde y)$ for $(g,\\tilde y)\\in G\\times\\tilde Y$ we can conclude that $(f_c\\circ\\sigma-f_Y\\circ\\rho\\circ\\sigma)(\\tau,\\tilde y)$ is uniformly bounded and a coboundary.\nHence also $(f_c-f_Y\\circ\\rho)(\\tau,y_c)$ is a coboundary, and the relative triviality of $f_Y(\\tau,y)$ with respect to $(f_\\alpha\\circ\\rho_\\alpha)(\\tau,y)$ can be verified as above for the cocycle $(f_c\\circ\\sigma)(\\tau,\\tilde y)$.\n\\end{proof}\n\n\\begin{proposition}\\label{prop:inc}\nThe cocycle $(f_Y-f_\\alpha\\circ\\rho_\\alpha)(\\tau,y)$ of the flow $(Y,T)$ can be extended to a cocycle $\\bar f((\\tau,t),y)$ of the $T\\times\\mathbb R$-flow $(Y,\\{\\psi^t\\circ\\tau:(\\tau,t)\\in T\\times\\mathbb R\\})$ so that\n\\begin{equation*}\n(f_Y-f_\\alpha\\circ\\rho_\\alpha)(\\tau,y)=\\bar f((\\tau,0),y)\n\\end{equation*}\nfor every $(\\tau,y)\\in T\\times Y$.\nWe put $L_\\tau (x_\\alpha,m)=(\\tau x_\\alpha,m)=\\psi^{-f_\\alpha(\\tau,x_\\alpha)}(\\tau (x_\\alpha,m))$ for $(x_\\alpha,m)\\in X_\\alpha\\times M=Y$.\nThen for arbitrary $\\bar x\\in X_\\alpha$ there exists a continuous function $\\bar b:Y\\longrightarrow\\mathbb R$ with $\\bar b(\\rho_\\alpha^{-1}(\\bar x))=\\{0\\}$ so that for every $(\\tau,y)\\in T\\times Y$ holds\n\\begin{equation}\\label{eq:tf}\n\\bar f((\\tau,-(f_\\alpha\\circ\\rho_\\alpha)(\\tau,y)),y)=\\bar b(\\psi^{-(f_\\alpha\\circ\\rho_\\alpha)(\\tau,y)} (\\tau y))-\\bar b(y) =\\bar b(L_\\tau y)-\\bar b(y) ,\n\\end{equation}\nand $(\\tau,y)\\mapsto\\bar f((\\tau,-(f_\\alpha\\circ\\rho_\\alpha)(\\tau,y)),y)$ is a topological coboundary of the distal flow $(Y,\\{L_\\tau:\\tau\\in T\\})$ with transfer function $\\bar b$.\nFor every $(x_\\alpha,m)\\in Y$ and $t\\in\\mathbb R$ holds\n\\begin{equation}\\label{eq:tF}\n\\bar f((\\mathbf 1_T,t),(x_\\alpha,m))= \\bar f((\\mathbf 1_T,t),(\\bar x,m))+\\bar b(x_\\alpha,\\phi^t(m))-\\bar b(x_\\alpha,m) .\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\nSince Lemma \\ref{lem:tilde_coc} can be applied to the cocycle $h=f_Y-f_\\alpha\\circ\\rho_\\alpha$, the group $G=\\{\\psi^t:t\\in\\mathbb R\\}\\subset\\textup{Aut}(Y,T)$, and the group homomorphism $\\varphi=\\textup{id}_\\mathbb R$, it provides the cocycle $\\bar f((\\tau,t),y)$.\nThe mapping $f'((\\tau,t),y)=\\bar f((\\tau,t-(f_\\alpha\\circ\\rho_\\alpha)(\\tau,y)),y)$ fulfils\n\\begin{eqnarray*}\nf'((\\tau,t),\\psi^{t'}(L_{\\tau'} y))+f'((\\tau',t'),y)=\\hspace{55mm}\\\\\n=\\bar f((\\tau,t-(f_\\alpha\\circ\\rho_\\alpha)(\\tau,\\psi^{t'}(L_{\\tau'} y))),\\psi^{t'}(L_{\\tau'} y))+\\bar f((\\tau',t'-(f_\\alpha\\circ\\rho_\\alpha)(\\tau',y)),y)=\\hspace{-7mm}\\\\\n=\\bar f((\\tau\\tau',t+t'-(f_\\alpha\\circ\\rho_\\alpha)(\\tau\\tau',y)),y)=f'((\\tau\\tau',t+t'),y)\n\\end{eqnarray*}\nand is thus a cocycle of the \\emph{minimal} flow $(Y,\\{\\psi^t\\circ L_\\tau:(\\tau,t)\\in T\\times\\mathbb R\\})$.\nNow let $(\\tau,y)\\in T\\times Y$ be arbitrary.\nBy equality (\\ref{eq:o_flow}) and the density of $\\widetilde{\\tau}_{f_\\alpha}$-transitive points in $X_\\alpha$, there exists a sequence $\\{(\\tau_k,y_k)\\}_{k\\geq 1}\\subset T\\times Y$ so that $y_k\\to\\tau y$, $\\tau_k y_k\\to\\psi^{-(f_\\alpha\\circ\\rho_\\alpha)(\\tau,y)}(\\tau y)$, and $(f_\\alpha\\circ\\rho_\\alpha)(\\tau_k,y_k)\\to -(f_\\alpha\\circ\\rho_\\alpha)(\\tau,y)$ as $k\\to\\infty$, and by equality (\\ref{eq:h_uni}) holds $(f_Y-f_\\alpha\\circ\\rho_\\alpha)(\\tau_k,y_k)\\to\\bar f((\\mathbf 1_T,-(f_\\alpha\\circ\\rho_\\alpha)(\\tau,y)),\\tau y)$.\nThus\n\\begin{equation*}\n(f_Y-f_\\alpha\\circ\\rho_\\alpha)(\\tau_k\\tau,\\tau^{-1} y_k)\\to\\bar f((\\mathbf 1_T,-(f_\\alpha\\circ\\rho_\\alpha)(\\tau,y)),\\tau y)+(f_Y-f_\\alpha\\circ\\rho_\\alpha)(\\tau,y)\n\\end{equation*}\nas $k\\to\\infty$, and this limit coincides with $\\bar f((\\tau,-(f_\\alpha\\circ\\rho_\\alpha)(\\tau,y)),y)=f'((\\tau,0),y)$ due to the cocycle identity for $\\bar f((\\tau,t),y)$.\nSince $f_Y(\\tau,y)$ is relatively trivial with respect to $(f_\\alpha\\circ\\rho_\\alpha)(\\tau,y)$, for every $\\varepsilon>0$ there exists a $\\delta>0$ so that for every $(\\tau',y')\\in T\\times Y$ with $d_Y(y',\\tau' y')<\\delta$ and $|(f_\\alpha\\circ\\rho_\\alpha)(\\tau',y')|<\\delta$ holds $|(f_Y-f_\\alpha\\circ\\rho_\\alpha)(\\tau',y')|<\\varepsilon$.\nFrom $\\tau^{-1} y_k\\to y$, $\\tau_k y_k\\to L_\\tau y$, and $(f_\\alpha\\circ\\rho_\\alpha)(\\tau_k\\tau,\\tau^{-1} y_k)\\to 0$ follows for every $(\\tau,y)\\in T\\times Y$ with $d_Y(y,L_\\tau(y))<\\delta$ that $f'((\\tau,0),y)<\\varepsilon$.\nFact \\ref{fact:GH} implies that the cocycle $(\\tau,y)\\mapsto f'((\\tau,0),y)$ of the distal flow $(Y,\\{L_\\tau:\\tau\\in T\\})$ is a coboundary on the $\\{L_\\tau:\\tau\\in T\\}$-orbit closure $X_\\alpha\\times\\{m\\}$ with transfer function $b_m:X_\\alpha\\longrightarrow\\mathbb R$ for every $m\\in M$.\nSince $\\delta>0$ is valid for all $\\{L_\\tau:\\tau\\in T\\}$-orbit closures, the transfer functions $\\{b_m:m\\in M\\}$ are uniformly equicontinuous.\nWe fix a point $\\bar x\\in X_\\alpha$ and obtain from the cocycle identity for all $(\\tau,t)\\in T\\times\\mathbb R$ and $(x_\\alpha,m)\\in Y$ that\n\\begin{eqnarray*}\nf_{\\bar x}((\\tau,t),(x_\\alpha,m))=f'((\\tau,t),(x_\\alpha,m))-f'((\\mathbf 1_T,t),(\\bar x,m))=\\\\\n=b_{\\phi^t(m)}(\\tau x_\\alpha)-b_{\\phi^t(m)}(\\bar x)-b_{m}(\\tau x_\\alpha)+b_m(\\bar x) .\n\\end{eqnarray*}\nThe function $f_{\\bar x}((\\tau,t),(x_\\alpha,m))$ is also a cocycle of $(Y,\\{\\psi^t\\circ L_\\tau:(\\tau,t)\\in T\\times\\mathbb R\\})$ and bounded on $T\\times\\mathbb R\\times Y$, hence a coboundary with a transfer function $\\bar b:Y\\longrightarrow\\mathbb R$ so that $\\bar b(\\rho_\\alpha^{-1}(\\bar x))=\\{0\\}$.\nNow equality (\\ref{eq:tf}) follows, and equality (\\ref{eq:tF}) follows from $f'((\\mathbf 1_T,t),(x_\\alpha,m))=\\bar f((\\mathbf 1_T,t),(x_\\alpha,m))$ for all $t\\in\\mathbb R$ and $(x_\\alpha,m)\\in Y$.\n\\end{proof}\n\nWith these prerequisites we can conclude the proof of our main result.\n\n\\begin{proof}[Proof of the structure theorem]\nWe let all  elements of the theorem and the flow $\\{\\psi^t:t\\in\\mathbb R\\}\\subset\\textup{Aut}(Y,\\{L_\\tau:\\tau\\in T\\})\\cap\\textup{Aut}(Y,T)$ be defined according to the Propositions \\ref{prop:max}, \\ref{prop:flow}, \\ref{prop:res}, and \\ref{prop:inc}.\nWe fix a point $\\bar x\\in X_\\alpha$ so that $(\\bar x,0)$ is transitive for $\\widetilde\\tau_{f_\\alpha}$ and $\\bar b(\\rho_\\alpha^{-1}(\\bar x))=\\{0\\}$.\nThen we define a cocycle $g(t,m)$ of the distal minimal flow $(M,\\{\\phi^t:t\\in\\mathbb R\\})$ by\n\\begin{equation*}\ng(t,m)=\\bar f((\\mathbf 1_T,t),(\\bar x,m)) \\quad\\textup{for all}\\enspace (t,m)\\in\\mathbb R\\times M.\n\\end{equation*}\nFrom equalities (\\ref{eq:tf}) and (\\ref{eq:tF}) follows for all $\\tau\\in T$ and $(x_\\alpha,m)\\in Y$ that\n\\begin{eqnarray*}\n(f_Y-f_\\alpha\\circ\\rho_\\alpha)(\\tau,(x_\\alpha,m))=\\bar f((\\tau,0),(x_\\alpha,m))=\\hspace{3cm}\\\\\n=\\bar f((\\mathbf 1_T,f_\\alpha(\\tau,x_\\alpha)),L_\\tau (x_\\alpha,m))+\\bar b(L_\\tau (x_\\alpha,m))-\\bar b(x_\\alpha,m)=\\\\\n=g(f_\\alpha(\\tau,x_\\alpha),m)+\\bar b\\circ \\tau(x_\\alpha,m)-\\bar b(x_\\alpha,m) .\n\\end{eqnarray*}\nHence equality (\\ref{eq:f_Y2}) holds for the cocycle $f_Y(\\tau,y)-\\bar b(\\tau y)+\\bar b(y)$ cohomologous to $f_Y(\\tau,y)$, and this cocycle will be substituted for $f_Y(\\tau,y)$ henceforth.\nFor every sequence $\\{(\\tau_k,x_k)\\}_{k\\geq 1}\\subset T\\times X$ with $(f_\\alpha\\circ\\pi_\\alpha)(\\tau_k,x_k)\\to 0$ it holds also that $(f_Y\\circ\\pi_Y)(\\tau_k,x_k)\\to 0$ as $k\\to\\infty$, and thus identity (\\ref{eq:p_X}) implies identity (\\ref{eq:fpy}).\n\nFor every ordinal $\\xi$ with $\\alpha\\leq\\xi\\leq\\eta$ we can apply the Propositions \\ref{prop:flow}, \\ref{prop:res}, and \\ref{prop:inc} to the distal minimal flow $(X_\\xi,T)$ and the cocycle $f_\\xi(\\tau,x_\\xi)$.\nWe obtain a factor $(Y_\\xi,T)=\\pi_{Y_\\xi}(X_\\xi,T)$ with $(X_\\alpha,T)=\\rho_\\alpha^\\xi(Y_\\xi,T)$, an $\\mathbb R$-flow $\\{\\psi_\\xi^t:t\\in\\mathbb R\\}\\subset\\textup{Aut}(Y_\\xi,T)$, a cocycle $f_{Y_\\xi}(\\tau,y_\\xi)$ of $(X_\\xi,T)$, and a cocycle $\\bar f_\\xi((\\tau,t),y_\\xi)$ of the flow $(Y_\\xi,T\\times\\mathbb R)$ extending the cocycle $(f_{Y_\\xi}-f_\\alpha\\circ\\rho^\\xi_\\alpha)(\\tau,y_\\xi)$.\nStriving for a contradiction to the maximality of the ordinal $\\alpha$ (cf. Proposition \\ref{prop:max}), we assume that the cocycle $(\\mathbbm 1+g)(t,m)$ of the minimal flow $(M,\\{\\phi^t:t\\in\\mathbb R\\})$ is recurrent so that the cocycle $\\bar f_\\eta((\\mathbf 1_T,t),y_\\eta)+t$ of the minimal flow $((\\rho^\\eta_\\alpha)^{-1}(\\bar x),\\{\\psi_\\eta^t:t\\in\\mathbb R\\})$ is also recurrent.\nWe let $\\beta$ be the minimal element of the non-empty set of ordinals\n\\begin{equation*}\n\\{\\alpha\\leq\\xi\\leq\\eta: \\bar f_\\xi((\\mathbf 1_T,t),y_\\xi)+t\\enspace\\textup{is a recurrent cocycle of}\\enspace ((\\rho_\\alpha^\\xi)^{-1}(\\bar x),\\{\\psi_\\xi^t:t\\in\\mathbb R\\})\\} ,\n\\end{equation*}\nwith $\\beta>\\alpha$ since $f_{Y_\\alpha}=f_\\alpha$ and $\\bar f_\\alpha((\\mathbf 1_T,t),y_\\alpha)\\equiv 0$.\nWe fix a point $\\bar x_\\beta\\in(\\pi_\\alpha^\\beta)^{-1}(\\bar x)$ so that $(\\pi_{Y_\\beta}(\\bar x_\\beta),0)$ is a recurrent point for the skew product extension of the flow $((\\rho_\\alpha^\\beta)^{-1}(\\bar x),\\{\\psi_\\beta^t:t\\in\\mathbb R\\})$ by the cocycle $\\bar f_\\beta((\\mathbf 1_T,t),y_\\beta)+t$.\nThen there exists a sequence $\\{\\bar\\tau_k\\}_{k\\geq 1}\\subset T$ with $f_\\alpha(\\bar\\tau_k,\\pi_\\alpha^\\beta(\\bar x_\\beta))\\to\\infty$ so that\n\\begin{equation*}\n\\bar f_\\beta((\\mathbf 1_T,f_\\alpha(\\bar\\tau_k,\\pi_\\alpha^\\beta(\\bar x_\\beta))),\\pi_{Y_\\beta}(\\bar x_\\beta))+f_\\alpha(\\bar\\tau_k,\\pi_\\alpha^\\beta(\\bar x_\\beta))=f_{Y_\\beta}(\\bar\\tau_k,\\pi_{Y_\\beta}(\\bar x_\\beta))\\to 0\n\\end{equation*}\nfor $k\\to\\infty$, and by the cohomology of the cocycles $f_\\beta(\\tau,x_\\beta)$ and $(f_{Y_\\beta}\\circ\\pi_{Y_\\beta})(\\tau,x_\\beta)$ the sequence $f_\\beta(\\bar\\tau_k,\\bar x_\\beta)$ is bounded.\nHence there exists a sequence $\\{k_l\\}_{l\\geq 1}\\subset\\mathbb N$ so that $f_\\alpha(\\bar\\tau_{k_{l+1}},\\pi_\\alpha^\\beta(\\bar\\tau_{k_l}\\bar x_\\beta))\\to\\infty$, $d_\\beta(\\bar\\tau_{k_{l+1}} \\bar x_\\beta,\\bar\\tau_{k_l}\\bar x_\\beta)\\to 0$, and $f_\\beta(\\bar\\tau_{k_l},\\bar x_\\beta)$ is convergent.\nThen the sequence $\\{(\\tau_l,x_l)=(\\bar\\tau_{k_{l+1}}(\\bar\\tau_{k_l})^{-1},\\bar\\tau_{k_l}\\bar x_\\beta)\\}_{l\\geq 1}\\subset T\\times X_\\beta$ fulfils $f_\\alpha(\\tau_l,\\pi_\\alpha^\\beta(x_l))\\to\\infty$, $d_\\beta(x_l,\\tau_l x_l)\\to 0$, and $f_\\beta(\\tau_l,x_l)\\to 0$ for $l\\to\\infty$.\nHowever, for every $\\alpha\\leq\\xi<\\beta$ holds\n\\begin{equation*}\n\\bar f_\\xi((\\mathbf 1_T,f_\\alpha(\\tau_l,\\pi_\\alpha^\\beta(x_l))),\\pi_{Y_\\xi}\\circ\\pi_\\xi^\\beta(x_l))+f_\\alpha(\\tau_l,\\pi_\\alpha^\\beta(x_l))=f_{Y_\\xi}(\\tau_l,\\pi_{Y_\\xi}\\circ\\pi_\\xi^\\beta(x_l))\\to\\infty\n\\end{equation*}\nfor $l\\to\\infty$.\nOtherwise, since $|\\bar f_\\xi((\\mathbf 1_T,t),(x_\\alpha,m_\\xi))-\\bar f_\\xi((\\mathbf 1_T,t),(\\bar x,m_\\xi))|$ is uniformly bounded for all $t\\in\\mathbb R$, $x_\\alpha\\in X_\\alpha$, and $m_\\xi\\in M_\\xi$ (cf. identity (\\ref{eq:tF})), there exists a non-trivial prolongation in the skew product of the minimal flow $((\\rho_\\alpha^\\xi)^{-1}(\\bar x),\\{\\psi_\\xi^t:t\\in\\mathbb R\\})$ and its cocycle $\\bar f_\\xi((\\mathbf 1_T,t),y_\\xi)+t$, which sufficient for its recurrence (cf. Lemma \\ref{lem:tr_coc}).\nTherefore also $f_\\xi(\\tau_l,\\pi_\\xi^\\beta(x_l))\\to\\infty$ as $l\\to\\infty$, and depending on the type of the ordinal $\\beta$ follows either from Lemma \\ref{lem:c_t} or Lemma \\ref{lem:lim} that $\\widetilde\\tau_{f_\\beta}$ is point transitive, in contradiction to the maximality of $\\alpha$.\n\\end{proof}\n\n\\begin{proof}[Proof of the structure of the topological Mackey action]\nIn the proof of the decomposition theorem it is verified that the topological Mackey actions for the cocycle $f(\\tau,x)$ of $(X,T)$ and the transient cocycle $(\\mathbbm 1+g)(t,m)$ of $(M,\\{\\phi^t:t\\in\\mathbb R\\})$ are topologically isomorphic.\nLet $\\{(M_\\xi,\\{\\phi_\\xi^t:t\\in\\mathbb R\\}):0\\leq\\xi\\leq\\theta\\}$ be the normal \\emph{I}-system for the distal minimal compact metric flow $(M,\\{\\phi^t:t\\in\\mathbb R\\})$ with the homomorphisms $\\sigma_\\xi:M\\longrightarrow M_\\xi$.\nFor every ordinal $0\\leq\\xi\\leq\\theta$ a cocycle $g_\\xi(t,m_\\xi)$ of $(M_\\xi,\\{\\phi_\\xi^t:t\\in\\mathbb R\\})$ is defined by a \\emph {RIM}.\nLet $\\beta$ be the minimal element of the set\n\\begin{equation*}\n\\theta\\in\\{0\\leq\\xi\\leq\\theta:(g-g_\\xi\\circ\\sigma_\\xi)(t,m)\\enspace\\textup{is a coboundary of}\\enspace(M,\\{\\phi^t:t\\in\\mathbb R\\})\\} .\n\\end{equation*}\nThe cocycle $(\\mathbbm 1+g_\\beta)(t,m_\\beta)$ is transient, since the cocycle $(\\mathbbm 1+g)(t,m)$ cohomologous to $(\\mathbbm 1+g_\\beta\\circ\\sigma_\\beta)(t,m)$ is transient.\nBy Lemma \\ref{lem:tr_coc} the right translation action $\\{R_b:b\\in\\mathbb R\\}$ acts minimally on the Fell compact space $D$ of orbits in $M_\\beta\\times\\mathbb R$.\nThe mapping $\\chi:M_\\beta\\longrightarrow D$ defined by $m_\\beta\\mapsto\\mathcal O_{\\phi_\\beta,(\\mathbbm 1+g_\\beta)}(m_\\beta,0)$ is Fell continuous, and for every $t\\in\\mathbb R$ holds $\\chi\\circ\\phi_\\beta^t(m_\\beta)=R_{(\\mathbbm 1+g_\\beta)(t,m_\\beta)}\\circ\\chi(m_\\beta)$.\nFor $\\beta=0$ the flow $(D,\\{R_b:b\\in\\mathbb R\\})$ is trivial and thus weakly mixing.\nIf $\\beta\\geq 1$, then $(D,\\{R_b:b\\in\\mathbb R\\})$ is a non-trivial minimal compact metric flow.\nIf it is not weakly mixing, then there exists a non-trivial equicontinuous factor $(D_1,\\{\\varphi^t:t\\in\\mathbb R\\})=\\nu(D,\\{R_b:b\\in\\mathbb R\\})$ with homomorphism $\\nu$ (cf. \\cite{KeRo}).\nWe shall use a generalised and relativised version of Theorem 1 in \\cite{Eg} to obtain a contradiction to the minimality of $\\beta$.\nSince $(D_1,\\{\\varphi^t:t\\in\\mathbb R\\})$ is a minimal and non-trivial flow, for each small enough $\\varepsilon>0$ holds $\\varphi^\\varepsilon(d_1)\\neq d_1$ for all $d_1\\in D_1$.\nWe shall verify as a sub-lemma that there are no sequences $\\{t_k\\}_{k\\geq 1}\\subset\\mathbb R$, $\\{m_k\\}_{k\\geq 1},\\{m'_k\\}_{k\\geq 1}\\subset M_\\beta$ so that $m_k\\to\\bar m$, $m'_k\\to\\bar m$, $\\phi_\\beta^{t_k}(m_k)\\to\\bar m'$, $\\phi_\\beta^{t_k}(m'_k)\\to\\bar m'$, and $g_\\beta(t_k,m'_k)-g_\\beta(t_k,m_k)\\to\\varepsilon$.\nIndeed,\n\\begin{gather*}\n\\nu\\circ\\chi\\circ\\phi_\\beta^{t_k}(m'_k)=\\varphi^{(\\mathbbm 1+g_\\beta)(t_k,m'_k)}\\circ\\nu\\circ\\chi(m'_k)\\to\\varphi^{\\varepsilon}(\\lim \\varphi^{(\\mathbbm 1+g_\\beta)(t_k,m_k)}\\circ\\nu\\circ\\chi(m'_k))\n\\end{gather*}\nand $\\lim\\varphi^{(\\mathbbm 1+g_\\beta)(t_k,m_k)}\\circ\\nu\\circ\\chi(m'_k)=\\lim\\varphi^{(\\mathbbm 1+g_\\beta)(t_k,m_k)}\\circ\\nu\\circ\\chi(m_k)=\\nu\\circ\\chi(\\bar m')$, by the equicontinuity of $(D_1,\\{\\varphi^t:t\\in\\mathbb R\\})$, imply that $\\nu\\circ\\chi(\\bar m')=\\varphi^{\\varepsilon}\\circ\\nu\\circ\\chi(\\bar m')$, which contradicts to the choice of $\\varepsilon$.\n\nIf $\\beta=\\gamma+1$ for some ordinal $0\\leq\\gamma<\\theta$, then the sub-lemma implies the uniform equicontinuity of the mapping $m_\\beta\\mapsto g_\\beta(t,m_\\beta)$ restricted on the $\\sigma_\\gamma^\\beta$-fibres and for all $t\\in\\mathbb R$.\nIndeed, otherwise we can find sequences as above with $\\sigma_\\gamma^\\beta(m_k)=\\sigma_\\gamma^\\beta(m'_k)$ for all $k\\geq 1$, and the condition on sufficiently small $\\varepsilon$ can be fulfilled by the connectedness of the $\\sigma_\\gamma^\\beta$-fibres.\nThus for all $t\\in\\mathbb R$ the cocycle $(g_\\beta-g_\\gamma\\circ\\sigma_\\gamma^\\beta)(t,m_\\beta)$ is uniformly equicontinuous and assumes zero on every connected $\\sigma_\\gamma^\\beta$-fibre.\nBy Fact \\ref{fact:GH} this is then a coboundary, in contradiction to the minimality of $\\beta$.\n\nIf $\\beta$ is a limit ordinal, then the sub-lemma applies to sequences so that for every ordinal $0\\leq\\xi<\\beta$ there exists an integer $k_\\xi\\geq 1$ with $\\sigma_\\xi^\\beta(m_k)=\\sigma_\\xi^\\beta(m'_k)$ for all $k\\geq k_\\xi$.\nHence there exists an ordinal $0\\leq\\zeta<\\beta$ so that $|g_\\beta(t,m_\\beta)-g_\\beta(t,m'_\\beta)|<\\varepsilon$ for all $m_\\beta,m'_\\beta\\in M_\\beta$ with $\\sigma_\\zeta^\\beta(m_\\beta)=\\sigma_\\zeta^\\beta(m'_\\beta)$ and for all $t\\in\\mathbb R$.\nIt follows that $(g_\\beta-g_\\zeta\\circ\\sigma_\\zeta^\\beta)(t,m_\\beta)$ is a coboundary, in contradiction to the minimality of $\\beta$.\n\nThe topological Mackey action of the transient cocycle $(\\mathbbm 1+g)(t,m)$ is topologically isomorphic to the topological Mackey action of the cohomologous cocycle $(\\mathbbm 1+g_\\beta\\circ\\sigma_\\beta)(t,m)$ (cf. the proof of  the decomposition theorem).\nThe weakly mixing flow $(D,\\{R_b:b\\in\\mathbb R\\})$ is a factor of the topological Mackey action of the cocycle $(\\mathbbm 1+g_\\beta\\circ\\sigma_\\beta)(t,m)$, since for every $(m,s)\\in M\\times\\mathbb R$ the mapping $\\sigma_\\beta\\times\\textup{id}_\\mathbb R$ maps the orbit $\\mathcal O_{\\phi,(\\mathbbm 1+g_\\beta\\circ\\sigma_\\beta)}(m,s)$ in $M\\times\\mathbb R$ to the orbit $\\mathcal O_{\\phi_\\beta,(\\mathbbm 1+g_\\beta)}(\\sigma_\\beta(m),s)$ in $M_\\beta\\times\\mathbb R$ continuously with respect to the Fell topologies.\nSuppose that there exists two distinct orbits $\\mathcal O, \\mathcal O'$ in $M\\times\\mathbb R$ within the same $\\sigma_\\beta\\times\\textup{id}_\\mathbb R$-fibre and $\\{t_k\\}_{k\\geq 1}\\subset\\mathbb R$ is a sequence with $R_{t_k}\\mathcal O\\to \\mathcal O''$ and $R_{t_k}\\mathcal O'\\to \\mathcal O''$.\nSince the mapping $t\\mapsto (\\mathbbm 1+g_\\beta)(t,m_\\beta)$ is onto $\\mathbb R$ for every $m_\\beta\\in M_\\beta$ (cf. Lemma \\ref{lem:tr_coc}), there exists a point $\\bar m\\in M_\\beta$ and distinct $m,m'\\in\\sigma_\\beta^{-1}(\\bar m)$ so that $(m,0)\\in\\mathcal O$ and $(m',0)\\in\\mathcal O'$.\nMoreover, for every integer $k\\geq 1$ we can select a real number $t'_k$ so that $(\\mathbbm 1+g_\\beta)(t'_k,\\bar m)=t_k$, and therefore $(\\phi^{t'_k}(m),0)\\in R_{t_k}\\mathcal O$ as well as $(\\phi^{t'_k}(m'),0)\\in R_{t_k}\\mathcal O'$.\nBy changing to a subsequence we can suppose that $\\phi^{t'_k}(m)\\to m_1$ and $\\phi^{t'_k}(m')\\to m_2$ as $k\\to\\infty$ with $m_1\\neq m_2$, by the distality of the flow $(M,\\{\\phi^t:t\\in\\mathbb R\\})$.\nHowever, since $(m_1,0),(m_2,0)\\in\\mathcal O''$, the point $(\\bar m,0)$ is a periodic point in $(D,\\{R_b:b\\in\\mathbb R\\})$ in contradiction to the transience of the cocycle $(\\mathbbm 1+g_\\beta)(t,m_\\beta)$.\nWe can conclude that $\\sigma_\\beta\\times\\textup{id}_\\mathbb R$ is a distal homomorphism of the topological Mackey action of the cocycle $(\\mathbbm 1+g_\\beta\\circ\\sigma_\\beta)(t,m)$ onto the weakly mixing flow $(D,\\{R_b:b\\in\\mathbb R\\})$.\n\\end{proof}\n\n\\textbf{Acknowledgement}: The author would like to thank Professor Jon Aaronson and Professor Eli Glasner for useful discussions and encouragement.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\nGender classification has an important role in modern society for surveillance or smart adaptation systems. It would be advantageous if a computer system or a machine could correctly classify an individual's gender. For example, a surveillance camera system of mall shoppers could be beneficial to know the gender of the customers to create a proper strategy, or a sale-man robot \\cite{ref0} could use an appropriate and smart approach to communicate with customers based on their gender.\n\n\nHuman gender, an active and promising area of research, can be classified using either a recorded voice \\cite{ref1}\\cite{ref2}\\cite{ref3} or face image \\cite{ref4}\\cite{ref5}\\cite{ref6}\\cite{ref7}\\cite{ref8}\\cite{ref9}\\cite{ref10}\\cite{ref10a}. SexNet \\cite{ref4} is an early system for gender classification using face images. The system uses the back-propagation algorithm of a neural network to train the gender classifier and obtains an error rate of 8.1\\%. Based on this encouraging result, the system demonstrates that automatic gender classification by computers is feasible. However, the use of voice and face features for gender classification has limitations when the objects are distant from the sensor because it is difficult to obtain a high-quality recorded voice or face image from a distance.\n\nMany psychological and medical experiments \\cite{ref11}\\cite{ref12} have indicated that humans and their gender can be recognized using their gait features. Therefore, gait features appear as alternative cues for resolving the recognition problem that occurs at long distances. Compared with other biometric features, gait information has particular advantages:\n\\begin{enumerate}\n\\item Easily obtainable from public areas and from a distance: Even when the subject is distant from the camera, we remain able to capture their gait information with an acceptable level of quality for specific tasks such as gait recognition and gender classification.\n\\item Uses simple instruments: Capturing the gait features requires only a simple conventional camera that can be placed anywhere in public areas such as banks, parking lots, and airports.\n\\item Does not require collaboration with the subjects: Gait features can be captured easily, even without the subject's permission. Although this is an advantage, it raises the issue of the right to privacy.\n\\item It is difficult to forge or falsify gait features: Gait features indicate the walking manner of a human, characterizing their physical capability. Mimicking the gait of other people is difficult.\n\\end{enumerate}\n\nHowever, gait-based systems such as gender classification and gait recognition share the same challenges as indicated in Fig.~\\ref{fig:1}. These challenges arise from the environment including the viewpoint of camera changes or the subject's physical characteristics such as carrying a backpack, wearing a heavy coat, or displaying signs of an injury. These factors change the subject's appearance leading to a significant effect on their gait information as they move \\cite{ref13}\\cite{ref14}.\n\n\n\\begin{figure}[!b]\n  \\centering\n  \\includegraphics[scale=0.35]{Fig1}\n\\caption{Challenges in gait analysis of humans: change in viewpoint (top row), carrying an item (middle row), wearing a coat (bottom row)}\n\\label{fig:1}      \n\\end{figure}\n\n\nThis paper proposes a novel gender classification method for use with an arbitrary viewpoint. To improve the performance, we present a method to remove areas with an attachment, such as a heavy coat or backpack. A general flowchart of the proposed method is presented in Fig. ~\\ref{fig:2}, which includes two major phases: training and testing. During the training phase, after the preprocessing step, the distance signal (DS) model, viewpoint (VP) model, and view-dependent gender classifier are built. Before building the VP model, the average gait image (AGI) and the lower portion of the average gait image (LAGI) are generated. During the testing phase, after the human detection and preprocessing step, the viewpoint of the current object is estimated using the current silhouette image and the VP model from the training phase. The attachment-area removal module is then used to eliminate unwanted areas such as backpacks or bags to obtain an attachment-free silhouette. Based on the estimated viewpoint of the current object, the corresponding classifier of that viewpoint (built during the training phase) is applied to the attachment-free silhouette to classify the object gender. The contributions of this paper are as follows:\n\\begin{itemize}\n\\item Building a VP model for viewpoint estimation, allowing the proposed method to estimate the viewing direction automatically.\n\\item Building a DS model for attachment-area removal to eliminate the noise generated from carried objects, which significantly degrades the performance of the system.\n\\item Building a viewpoint-dependent gender classifier using an SVM \\cite{ref15} that allows the algorithm to function from any viewpoint.\n\\end{itemize}\n\n\\begin{figure}[!b]\n\\centering\n\\includegraphics[scale=0.25]{Fig2}\n\\caption{Flowchart of proposed gender-classification process}\n\\label{fig:2}\n\\end{figure}\n\nThis paper is organized as follows. Section 2 discusses related works. Section 3 introduces the proposed method including the training and testing phases. The details of the key modules, such as VP modeling and estimation, DS modeling, viewpoint-dependent classifier building, and attachment-area removal are discussed in this section. A pseudo-code of the overall flowchart is also presented in this section. Section 4 presents the experimental results on public datasets. Finally, Section 5 concludes this paper and provides some areas for future work.\n\n\\section{Related works}\nGait-based recognition techniques can be divided into two categories, marker-based and markerless methods. Using markers, in an early work, Kozlowski and Cutting \\cite{ref16}\\cite{ref17} attempted to attach a point-light display (marker-based method) to a human body to extract the gait information. With this system, a human observer can determine a subject's gender based on the signals obtained with an acceptable level of accuracy (63\\%). However, to capture the gait information, the subject is required to wear a swimsuit and special devices, which is inconvenient, unfriendly, and impractical in real circumstances.\n\nToday, owing to technical innovations in camera and sensor development, the human gait can be easily obtained without a point-light display, leading to the development of markerless methods. The markerless-based methods for gait recognition can be classified into the model and appearance-based approaches. Such categorization can also be used for gender classification.\n\nIn the model-based methods \\cite{ref18}\\cite{ref19}\\cite{ref20}, the human body is divided into various parts, the structures of which are then fitted using primitive shapes such as ellipses, rectangles, and cylinders. Then, the gait feature is encoded using the parameters of the primitive shapes to measure the time-varying motion of the subject. In \\cite{ref18}, L. Lee et al. divide a human silhouette into seven different parts corresponding to the head and shoulder region, the front of the torso, back of the torso, front thigh, back thigh, front calf and foot, and back calf and foot. They then use ellipses to fit the model and capture the parameters of the ellipses such as the mean, standard deviation, orientation, and magnitude of the major components as feature vectors for classification. Although such methods are robust to noise and occlusions, they typically require a relatively high computational cost.\n\nAppearance-based methods \\cite{ref21}\\cite{ref22}\\cite{ref23}\\cite{ref24}\\cite{ref25}\\cite{ref26} analyze the spatio-temporal shape and dynamic motion characteristics of the silhouette in a gait sequence without using a human body model. A gait energy image (GEI) \\cite{ref21} is frequently used to encode the gait features because it includes both static (body shape) and dynamic information (arm swings and leg movements). The GEI feature is defined as the average frame of the subject in the gait cycle. Compared to the model-based methods, the appearance-based methods are considerably faster. In \\cite{ref23}, instead of modeling the silhouette, Shiqi Yu et al. calculate the GEI and use it to create the seven-part model defined in \\cite{ref17}. Because the contribution of each part to the gender classification varies, the authors assign different weights to the parts based on their experiments. Such methods obtain highly accurate classification rates (approximately 95\\%). However, they were developed to the only function on a side view, making it inappropriate to apply in real applications.\n\nIn \\cite{ref13}\\cite{ref14}\\cite{ref27}\\cite{ref28}\\cite{ref29}\\cite{ref30}\\cite{ref31}\\cite{ref32}\\cite{ref47}, the researchers attempted to resolve gender classification from multiple viewpoints. In \\cite{ref31}\\cite{ref32}, De Zhang et al. build an invariant classifier by combining the GEIs of different viewpoints into a single third-order tensor. They then use multiple linear principal component analysis (PCA) to reduce the dimensions and apply a support vector machine (SVM) to create a discriminative gender classifier. From another perspective, Kale et al. \\cite{ref33} use complicated equations from the structure of motion \\cite{ref34} to eliminate the viewpoint effect by synthesizing the side view from other viewpoints. A final recognition task is conducted on the synthesized data. Issac et al. \\cite{ref47} propose a method to delineate the gait instance as a sequence of poses or frames based on the fact that humans tend to assume certain poses at each part of a gait cycle. The gender of each frame is predicted, and the gender decision of a sequence is then made using majority voting. However, none of the previous works considers solving the problem of a subject carrying an item or wearing a heavy coat, which are common situations in real applications that can significantly degrade the classification rate.\n\n\\section{Proposed method}\n\\label{sec:3}\n\n\\subsection{Dataset and preprocessing step}\n\\label{sec:31}\nThis paper proposes a method for gender classification from an arbitrary viewpoint, and therefore, a dataset with multiple camera views is required. For this purpose, the CASIA gait Dataset B \\cite{ref13}\\cite{ref14} was utilized throughout this study for illustrative and experimental purposes.\n\nA person is first detected using a histogram of oriented gradient (HOG) \\cite{ref35}. During the preprocessing step, a classic background subtraction \\cite{ref36} is then applied to obtain a person's silhouette. Because a person's size changes from frame to frame, it is necessary to normalize the human bounding box before the training process. Assume in frame $I_t$ at time $t$, that a human is detected with a bounding box $B_t$; denote their silhouette obtained from the background subtraction as $S_t$. To register the silhouette image, the center of the silhouette $P_t$ (reference point) at time t is computed as:\n\\begin{equation}\n\\label{eq1}\nP_t(x_0, y_0) = \\left(\\frac{M_{10}}{M_{00}}, \\frac{M_{01}}{M_{00}}\\right)\n\\end{equation}\nwhere $M_{ij}$ are the raw moments of the binary silhouette image defined by $M_{ij}=\\sum_{x}\\sum_{y}x^iy^jS_{t}(x,y)$.\n\nThe silhouette image $S$ is resized to the fixed height $h$ to maintain the human ratio scale. The resized image is then zero padded or cropped on both sides (left and right sides) to ensure that the silhouette image has the predefined width $w$.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=1\\textwidth]{Fig3}\n\\caption{Example AGIs: models from different viewpoints}\n\\label{fig:3}\n\\end{figure*}\n\n\n\\subsection{Viewpoint modeling and estimation}\n\\label{sec:32}\n\\subsubsection{Viewpoint modeling}\n\\label{sec: 321}\nA person can move in an arbitrary direction under real circumstances. Therefore, in this paper, rather than using GEI \\cite{ref21} as a representation of a gait feature for classification, AGI is defined, as shown in Fig.~\\ref{fig:3}. The main difference between GEI and AGI is that the gait cycle information, which must be calculated into a GEI, is not required in an AGI. Furthermore, applying the gait cycle during the feature extraction step makes the entire algorithm inflexible because the gait cycle can be calculated accurately only when the person is captured from a side view, which is impractical under real circumstances. With the proposed method, all models are trained based on each viewpoint independently. Subsequently, we describe the details required for training for viewpoint $\\alpha$ using the data on that viewpoint. The AGI is defined as:\n\\begin{equation}\n\\label{eq2}\nAGI^{\\alpha}(x, y)=\\frac{1}{T}\\sum_{t=1}^{T}S_t^\\alpha(x,y)\n\\end{equation}\nwhere $T$, gait period, is defined adaptively using the video frame rate $f$ and approximate gait cycle time $\\mu$ as $T = \\mu*f$. According to \\cite{ref37}\\cite{ref38}, when the frame rate $f$ is 25 frames\/s, the value of the gait cycle time $\\mu$ must be 0.6 seconds to capture the most informative gait features; thus, $T = 0.6*f$ is used in the proposed method; $S_t^\\alpha$ is the silhouette image at time $t$ with the viewpoint $\\alpha$.\n\nDefining $\\gamma_k^\\alpha$ as $\\gamma_k^\\alpha=\\left\\{{AGI}_1^\\alpha, {AGI}_2^\\alpha, ..., {AGI}_n^\\alpha\\right\\}_k$, is the feature vector of subject $k$ in a viewpoint $\\alpha$, $\\alpha=\\overline{1, \\nu}$ and $k=\\overline{1,N}$, where $\\nu$ and $N$ are number of viewpoints and number of subjects (training samples), respectively. In fact, for the training step, a greater number of training samples $N$ is preferable. The corresponding label of $\\gamma_k^\\alpha$, denoted as set $L_k^\\alpha=\\{y_1^\\alpha, y_2^\\alpha, ..., y_n^\\alpha\\}_k, k=\\overline{1, N}; \\alpha=\\overline{1,\\nu}; y_i^\\alpha\\in\\{-1,1\\}$, is used to indicate the gender (\"-1\" for female and \"1\" for male).\n\nTo estimate the viewpoint for the input during the testing phase, we construct a viewpoint model $D$. This viewpoint model includes the viewpoint templates of an individual view. The viewpoint template is calculated as the average silhouette of all sequences from the $\\alpha$-th viewpoint. As observed, the viewpoint is clearly distinguished in the lower part of the silhouette and therefore, the $\\alpha$-th viewpoint template, denoted as ${LAGI}^\\alpha$, is extracted as the lower part of the average silhouette denoted as $LPS^\\alpha$ with a height of 0.715$h$ to $h$, as suggested by \\cite{ref39}, where $h$ is the height of the silhouette:\n\n\\begin{equation}\n\\label{eq3}\n{LPS}^\\alpha(x,y)=S^\\alpha(x,y), x=\\overline{0.715h, h}, y=\\overline{1, w}\n\\end{equation}\n\\begin{equation}\n\\label{eq4}\n{LAGI}^\\alpha(x,y) = \\frac{1}{N}\\sum_{t=1}^N{LPS_t^\\alpha(x,y)}\n\\end{equation}\n\nThe viewpoint model is then denoted as $D_{Low}=D=\\{{LAGI}^0,{LAGI}^1, \u2026,{LAGI}^\\nu \\}$, where $\\nu$ is the number of viewpoints. This viewpoint model $D_{Low}$ is used to estimate the viewpoint of a person walking during the testing phase.\n\n\\subsubsection{Viewpoint estimation}\nWith this method, the attachment-area removal module and gender classifier are dependent on the viewpoint; thus, the viewpoint is first estimated. During the testing phase, given the sequences of the silhouettes, the average gait image of the current walking subject, ${AGI}^c$, is calculated using Eq.~\\ref{eq2}. Then, ${LAGI}^c$  is extracted from the lower part of ${AGI}^c$  based on the size given in Eq.~\\ref{eq3}, rather than recalculating $LAGI$, as during the training phase.\n\nTo obtain viewpoint $\\alpha$ of the current walking subject, ${LAGI}^c$  is matched with each viewpoint template in viewpoint model $D_{Low}$ using the Euclidean distance. The least distance is then selected for the viewpoint estimation, as indicated in Fig. ~\\ref{fig:4}.\n\\begin{equation}\n\\label{eq5}\n\\alpha=\\min_j||{LAGI}^c-{LAGI}^j||_2, j=\\overline{0,\\nu}\n\\end{equation}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=1\\textwidth]{Fig4}\n\\caption{Viewpoint model of 11 viewpoint templates where ${LAGI}^c$ is matched with the first template}\n\\label{fig:4}\n\\end{figure*}\n\n\n\n\\subsection{Distance signal modeling and attachment removal}\n\\subsubsection{Distance modeling}\nIn real applications, it is common to view people moving with attached objects such as bags or backpacks; similarly, their appearance can be significantly changed when wearing a heavy coat. The added area resulting from a held item or worn coat contributes nothing to the result of the gender classification. In actuality, these factors negatively influence the results of the classification. In this section, a distance signal (DS) model of humans under normal walking conditions (not holding anything and wearing thin clothes) from different viewpoints is proposed for removing these redundant attachments.\n\nGiven a set of silhouettes in movement direction $\\alpha$ , for each silhouette, a distance signal is built. Considering the mass reference point calculated by Eq. ~\\ref{eq1}, each point $P_i$  on the silhouette boundary is represented in polar coordinates by two parameters, $d_i$ and $\\theta_i$, which indicate the distance from the point $P_i$ to the reference point $P$, and the angle formed by the line connecting the point to the reference point ${PP}_i$ with the horizon ${PP}_h$, respectively. \n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=1\\textwidth]{Fig5}\n\\caption{Example of distance signal (a) and distance signal model for different viewpoints (b)}\n\\label{fig:5}\n\\end{figure*}\n\n\n\\begin{equation}\n\\label{eq6}\nd_i=||P-P_i||_2\n\\end{equation}\n\n\\begin{equation}\n\\label{eq7}\n\\theta_i=\\arccos\\left(\\frac{\\overrightarrow{PP_i}*\\overrightarrow{PP_h}}{|\\overrightarrow{PP_i}||\\overrightarrow{PP_h}|}\\right)\n\\end{equation}\nwhere * is the dot product between two vectors and the value of angle $\\theta$ varies from $0^o$ to $360^o$ computed counterclockwise; $P_i$ and $P_h$ are depicted in Fig. \\ref{fig:5}a (left). The DS signal is then constructed by continuously concatenating these parameters from $P_h$ counterclockwise to define the signal presented in Fig. \\ref{fig:5}a (right). After building these DS signals for viewpoint $\\alpha$, denoted by ${DS}^\\alpha=\\{{DS}_1^\\alpha,\u2026,{DS}_n^\\alpha\\}$, the DS model for viewpoint $\\alpha$ is constructed using two curves, $MaDS^\\alpha$ and $MiDS^\\alpha$, which are defined as:\n\n\\begin{equation}\n\\label{eq8}\nMaDS^\\alpha=\\{d_i^{max}, \\theta_i\\}, \\text{where } d_i^{max} = \\max\\limits_{k=\\overline{1,N}}\\{d_k|\\theta_k\\}\n\\end{equation}\n\n\\begin{equation}\n\\label{eq9}\nMiDS^\\alpha=\\{d_i^{min}, \\theta_i\\}, \\text{where } d_i^{min}=\\min\\limits_{k=\\overline{1,N}}\\{d_k|\\theta_k\\}\n\\end{equation}\n\nThe distance signals are smoothed using the moving average technique with the number $n_{avg}=3$ before calculating the DS model. Fig. \\ref{fig:5}b illustrates the DS model for 11 viewpoints in our experiments.\n\n\n\\subsubsection{Attachment-area removal}\nDuring the testing phase, the DS signal of the silhouette of the current subject $DS^c$ is calculated in the manner described in the section 3.3.1. Given the viewpoint estimated using the viewpoint estimation module, the current $DS^c$ is projected to the corresponding viewpoint DS model. The current $DS^c$  is modified using the following rule to eliminate any attachments, if they exist:\n\n\\begin{equation}\n\\label{eq10}\nDS^c=\\{d_i^c, \\theta_i\\} \\text{where } d_i^c= \\begin{cases} d_i^c & \\text{if }  d_i^c \\leq d_i^{max} \\\\\nd_i^{min} & \\text{ otherwise}\n\\end{cases} \n\\end{equation}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=1\\textwidth]{Fig6}\n\\caption{Example of original silhouette, DS of the silhouette, corrected DS of the silhouette, and silhouette reconstruction of the current subject from a side view}\n\\label{fig:6}\n\\end{figure*}\n\n\nFig. \\ref{fig:6} illustrates this process. In the figure, an example of a side-view silhouette input with an attachment is provided because in this view the attachment can be seen most clearly. Our goal is to remove the attachment from the human silhouette, therefore, the human silhouette is divided into three parts, as shown in Fig. \\ref{fig:6}a. The first part consists of head and shoulder with a height of 0 to 0.17$h$, as suggested in \\cite{ref45}\\cite{ref46}, where $h$ is the height of human silhouette. The second part includes the human torso and thigh with a height of 0.17$h$ to 0.715$h$. Finally, the last part is human calf with a height of 0.715$h$ to $h$, as suggested in \\cite{ref39}. The red vertical lines (Fig. 6b, c, d, e ) are drawn to separate these three parts from the human silhouette. Firstly, the input silhouette is converted into a distance signal $DS^c$, presented as violet curve in Fig. \\ref{fig:6}b. The white curves are the maximum distance signal $MaDS^\\alpha$ (upper) and minimum distance signal $MiDS^\\alpha$ (lower) from the DS model for a specific viewpoint ($\\alpha=90^o$). \n\n\n\nWhen human carrying a backpack or bag, the appearance of torso thigh part is\nchanged due to the attachment, thus only this part is taken into account for\ncorrection. As we observed from the experiments, a $DS^c$ with a value less than\n$MiDS^\\alpha$ is the noise from an imperfect background subtraction. A $DS^c$\nwith a value greater than $MaDS^\\alpha$ is considered as the attachment area\nfrom a subject carrying an item while walking. Using the DS model from\nSection 3.3.1, the attachment area can therefore be removed. The corrected\nversion of $DS^c$ is obtained by replacing the violated signal with the \ncorresponding values of the $MiDS^\\alpha$ curve at Point A, as shown in Fig. \n\\ref{fig:6}c. To avoid the problem of strict change in the resulted signal, we \ncontinue to look from the point A to point B in Fig. \\ref{fig:6}c to find the \npoint that has the smallest vertical distance between $MiDS^\\alpha$ and the \nresulted signal (point C). The segment of the resulted signal from A to C is again \nreplaced by $MiDS^\\alpha$, as presented in Fig. \\ref{fig:6}d. The signal in Fig. \n\\ref{fig:6}e is obtained by smoothing using the average filter $\\frac{1}{5}[1 1 1 1 1]$. The same process is applied to all segments of torso\/thigh part of the $DS^c$ curve. Finally, the corrected version of the $DS^c$ is used to reconstruct the silhouette of attachment-free, as shown in Fig. \\ref{fig:6}f. The updated version of the silhouette is then used to recalculate the AGI for gender classification. \n\n\n\\subsection{Gender classifier building}\nSVM \\cite{ref15} is a superior tool for a binary classification problem regarding minimizing the classification error and maximizing the margin between the two classes. Because gender classification is a binary classification task, a standard SVM with a linear kernel was selected to train the view-dependent classifiers. For solving a constrained quadratic optimization problem, we set the maximum number of iteration to 100.\n\nTo create the viewpoint-dependent classifier, the feature sets $\\gamma^\\alpha=\\{\\gamma_1^\\alpha,\\gamma_2^\\alpha, \u2026,\\gamma_N^\\alpha\\}$ and its corresponding labels $L^\\alpha=\\{L_1^\\alpha,L_2^\\alpha, \u2026,L_N^\\alpha\\}$ are used as inputs for the linear SVM. The $\\alpha$-th viewpoint classifier, obtained by using the SVM, is denoted as  $C_{gen}^\\alpha$. The multiple-view classifier is a collection of different viewpoint-dependent classifiers, which is denoted as $C_{gen}={C_{gen}^\\alpha}$ and $\\alpha=\\overline{1, \\nu}$. \n\nIn the testing phase, the viewpoint is estimated as discussed in section 3.2.2. Based on the estimated viewpoint $\\alpha$, the corresponding classifier $C_{gen}^\\alpha$ is automatically selected from  $C_{gen}$ to predict the person's gender in a current frame.\n\nThe algorithms 1 and 2 give a more detail description of the proposed method using pseudo-code in which all notations described above are used.\n\n\\begin{algorithm*}\n\\caption{The training phase}\n\\begin{algorithmic}[1]\n\\STATE\\textbf{Input}: $video\\texttt{\\_}sequences$ in the $training\\texttt{\\_}samples$ of male and female in all viewpoints \n\\STATE \\textbf{Output}: VP model, DS model, and $C_{gen}$ classifier\n\n\\FORALL {$\\alpha$ \\textbf{in} $views$}\n\t\\FORALL {$video\\texttt{\\_}sequence\\; k$ \\textbf{in} $training\\texttt{\\_}samples$ of viewpoint $\\alpha$ }\n\t\t\\FORALL {frame $t$ \\textbf{in} $video\\texttt{\\_}sequence$}\n\t\t\t\\STATE Human detection\n\t\t\t\\IF {no human detected}\n\t\t\t\t\\STATE Skip to the next frame\n\t\t\t\\ENDIF\n\t\t\t\\STATE Preprocessing to get the normalized silhouette $S^\\alpha$\n\t\t\t\\STATE Extract low part of the silhouette $S^\\alpha$ by Eq.~\\ref{eq3} and accumulate to $LPS_{k,t}^\\alpha$\n\t\t\t\\STATE Detect contour of a normalized silhouette $S^\\alpha$\n\t\t\t\\STATE Calculate  $d_i$ and $\\theta_i$ by Eq.~\\ref{eq6}, Eq.~\\ref{eq7}\n\t\t\t\\STATE Calculate $AGI^\\alpha$ and assign its label $y^\\alpha$ by Eq.~\\ref{eq2}\n\t\t\t\\STATE Append $AGI^\\alpha$ to vector $\\gamma_k^\\alpha$\n\t\t\t\\STATE Append $y^\\alpha$ to vector $L_k^\\alpha$\n\t\t\\ENDFOR\n\t\\ENDFOR\n\t\\STATE Calculate the $LAGI^\\alpha$ using $LPS_{t,f}^\\alpha$ by Eq.~\\ref{eq4} for VP model\n\t\\STATE Calculate $MaDS^\\alpha$ and $MiDS^\\alpha$ using $d_i$ and $\\theta_i$ by Eq.~\\ref{eq8}, Eq.~\\ref{eq9} for DS model\n\t\\STATE Train view-dependent classifier $C_{gen}^\\alpha$ using $\\gamma^\\alpha$ and $L^\\alpha$ as inputs of SVM\n\\ENDFOR\n\\end{algorithmic}\n\\end{algorithm*}\n\n\n\\begin{algorithm*}\n\\caption{The testing phase}\n\\begin{algorithmic}[1]\n\\STATE \\textbf{Input}: $video\\texttt{\\_}sequence$ of a person in an unknown viewing angle\n\\STATE \\textbf{Output}: Gender information\n\\STATE Initialize $counter = 0$\n\\STATE Initialize empty vector $v$\n\\FORALL {frame $t$ \\textbf{in} $video\\texttt{\\_}sequence$}\n\t\\STATE Human detection\n\t\\IF {human detected}\n\t\t\\STATE Increase $counter$ by 1\n\t\\ELSE\n\t\t\\STATE $counter=0$\n\t\t\\STATE Skip to the next frame\n\t\\ENDIF\n\t\\STATE Preprocessing to get the normalized silhouette $S$\n\t\\STATE Append $S$ to the end of vector $v$\n\t\\IF {$counter\\geq15$}\n\t\t\\STATE Calculate the $AGI$ using a vector of silhouette $v$\n\t\t\\STATE Extract low part of average gait image of the current frame $LAGI^c$ from $AGI$\n\t\t\\STATE Estimate the viewpoint $\\alpha$ using $LAGI^c$ and VP model by Eq.~\\ref{eq5}\n\t\t\\STATE Remove the attachment area using estimated viewpoint $\\alpha$ and DS model by Eq.~\\ref{eq10}\n\t\t\\STATE Reconstruct silhouette $S\\rightarrow S$' and update $AGI\\rightarrow AGI$'\n\t\t\\STATE Predict the gender in current frame using the updated $AGI$' and the estimated viewpoint $\\alpha$\n\t\t\\STATE Remove the first element of $v$\n\t\\ENDIF\n\\ENDFOR\n\\end{algorithmic}\n\\end{algorithm*}\n\n\\section{Experimental results}\n\\subsection{Experimental dataset}\nThe CASIA Dataset B \\cite{ref13}\\cite{ref14} addresses our requirements of multiple camera views because it includes sequences of various people from 11 viewpoints (from $0^o$ to $180^o$) under different walking conditions such as walking normally, carrying a backpack, and wearing a coat. The CASIA Dataset B captures sequences of 124 individual people (31 females and 93 males). Each person is captured ten times to create ten different sequences including six sequences under normal walking conditions, two backpack-carrying sequences, and two coat-wearing sequences. Table~\\ref{tab:1} summarizes the information of the CASIA Dataset B.\n\n\n\n\n\\begin{table}[!b]\n\\centering\n\\caption{Summary of CASIA Dataset B}\n\\label{tab:1}      \n\\begin{tabular}{lll}\n\\hline\\noalign{\\smallskip}\n\\text{Walking condition} & \\text{\\#subjects} & \\text{\\#sequences}  \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n\\text{Normal walking} & 6 & $6\\times124\\times11$ \\\\\n\\text{Carrying a bag} & 2 & $2\\times124\\times11$ \\\\\n\\text{Wearing a coat} & 2 & $2\\times124\\times11$ \\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{table}\n\nThe CASIA Dataset B includes background subtraction and thus, in the proposed system we are only required to resize and center the silhouette to the same size (144$\\times$144). For the AGI calculation, we must accumulate fifteen frames; it requires approximately 0.6 seconds to obtain the first gender-classification result when the frame rate is 25 fps, which can be considered a system delay.\n\nWe used the CASIA Dataset B for both training and testing using the same protocol as in \\cite{ref32}, which uses n-fold cross-validation. With this protocol, all 31 females were selected; 31 males were selected randomly from the CASIA Dataset B owing to a bias in the number of males in the dataset. The 31 females and 31 males were then grouped into 31 disjoint sets consisting of one female and one male. To create viewpoint-dependent classifiers, we use 30 sets for training. The remaining sets were used to test the system accuracy. The training and testing phases were repeated 31 times; the averages of the correct classification rate are listed for all experiments.\n\\subsection{Viewpoint-dependent classifiers test}\nThis test was used to validate the performance of only viewpoint-dependent classifiers under the assumption that the viewpoint was given. We conducted the test for both correct and incorrect viewpoint classifiers with respect to a specific viewpoint to observe the effect of viewpoint changes on the gender classification. Table \\ref{tab:2} displays the correct classification rates ($CCRs$) when using the corresponding classifier and a non-corresponding classifier (the viewpoint is given). The $CCR$ is defined as:\n\\begin{equation}\n\\label{eq11}\nCCR=\\frac{TP+TN}{N}\n\\end{equation}\nwhere $TP$ is the true positive referring to the cases in which the system correctly classifies positive samples (male to male), $TN$ is the true negative referring to the cases in which the system correctly classifies negative samples (female to female), and $N$ is the total number of samples. In these experiments, the male samples are labeled as 1 (positive samples) and the female samples are labeled as -1 (negative samples). As indicated in Table \\ref{tab:2}, applying a proper classifier for a specific viewpoint provides higher $CCRs$ (97.6\\% $\\pm$ 0.881) (for further description see Table \\ref{tab:2}).\n\n\\begin{table*}\n\\centering\n\\caption{CCRs (\\%) of viewpoint-dependent classifiers for specific viewing angle under normal walking conditions}\n\\label{tab:2}      \n\\begin{tabular}{llllllllllll}\n\\hline\\noalign{\\smallskip}\n\\multirow{2}{*}{\\text{Ground-truth viewpoint}} & \\multicolumn{11}{c}{Classifier} \\\\\n& $0^o$ & $18^o$ & $36^o$ & $54^o$ & $72^o$ & $90^o$ & $108^o$ & $126^o$ & $144^o$ & $162^o$ & $180^o$  \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n$0^o$ &97.6&96.5&94.1&89.6&89.9&82.3&87.2&93.8&94.6&94.5&95.8 \\\\\n$18^o$&97.5&98.6&96.8&93.4&89.1&85.4&88.6&92.4&93.7&95.2&96.4  \\\\\n$36^o$&95.3&96.6&97.4&95.8&94.2&93.4&93.9&94.5&95.7&95.7&94.2  \\\\\n$54^o$&92.2&95.3&96.1&96.6&95.8&94.6&93.4&95.0&95.5&94.6&93.1 \\\\\n$72^o$&92.3&93.2&94.1&95.4&96.1&94.8&94.5&95.2&93.0&92.8&92.4 \\\\\n$90^o$&90.0&93.1&95.3&95.5&95.7&98.8&96.9&95.7&94.1&92.5&91.1 \\\\\n$108^o$&91.4&92.7&95.4&96.1&96.5&97.0&97.3&96.2&95.5&92.4&93.2 \\\\\n$126^o$&92.5&94.3&96.7&95.7&95.3&94.1&93.4&96.8&94.5&94.7&94.5 \\\\\n$144^o$&95.7&96.8&97.8&95.8&93.2&93.4&93.9&94.8&97.5&95.5&94.5 \\\\\n$162^o$&95.5&96.3&95.8&93.4&91.3&90.4&92.6&94.4&95.4&98.3&96.6 \\\\\n$180^o$&96.8&97.6&95.8&92.8&91.5&91.4&92.6&92.3&93.5&95.3&98.5 \\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{table*}\n\n\n\\begin{table*}\n\\centering\n\\caption{CCRs (\\%) of viewpoint-dependent classifiers of specific viewing angle when carrying a bag}\n\\label{tab:3}      \n\\begin{tabular}{llllllllllll}\n\\hline\\noalign{\\smallskip}\n\\multirow{2}{*}{\\text{Ground-truth viewpoint}} & \\multicolumn{11}{c}{Classifier} \\\\\n & $0^o$ & $18^o$ & $36^o$ & $54^o$ & $72^o$ & $90^o$ & $108^o$ & $126^o$ & $144^o$ & $162^o$ & $180^o$  \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n$0^o$&94.3&91.6&89.7&84.8&81.5&79.6&80.3&83.9&85.2&89.9&92.6 \\\\\n$18^o$&92.8&94.4&93.5&89.9&83.1&80.1&82.8&86.4&85.4&92.2&91.7  \\\\\n$36^o$&89.8&92.1&93.7&90.7&85.2&82.0&88.8&90.2&86.4&90.5&89.1  \\\\\n$54^o$&88.0&88.3&90.1&91.2&89.1&84.5&89.1&90.6&87.9&88.3&86.4 \\\\\n$72^o$&88.1&89.4&89.5&90.4&90.6&86.2&89.4&89.1&89.3&88.8&84.4 \\\\\n$90^o$&82.3&82.8&83.1&85.5&86.4&87.4&87.1&85.3&85.1&83.6&82.1 \\\\\n$108^o$&82.2&82.4&83.9&84.2&85.5&86.7&89.8&87.4&87.0&85.4&83.7 \\\\\n$126^o$&87.4&87.3&88.3&90.7&83.3&85.5&88.1&91.2&90.5&89.7&88.5 \\\\\n$144^o$&88.7&89.3&90.6&90.1&82.2&83.4&87.3&90.2&91.4&89.8&88.4 \\\\\n$162^o$&91.4&91.2&90.8&89.4&81.1&82.1&85.6&89.6&90.7&93.1&92.2 \\\\\n$180^o$&93.2&92.5&91.0&90.7&80.5&80.6&84.1&87.1&89.1&91.4&94.8 \\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{table*}\n\n\\begin{table*}\n\\centering\n\\caption{CCRs (\\%) of viewpoint-dependent classifiers of specific viewing angle when wearing a coat }\n\\label{tab:4}      \n\\begin{tabular}{llllllllllll}\n\\hline\\noalign{\\smallskip}\n\\multirow{2}{*}{\\text{Ground-truth viewpoint}}& \\multicolumn{11}{c}{Classifier} \\\\\n& $0^o$ & $18^o$ & $36^o$ & $54^o$ & $72^o$ & $90^o$ & $108^o$ & $126^o$ & $144^o$ & $162^o$ & $180^o$  \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n$0^o$&92.2&91.5&89.2&87.4&86.1&85.5&87.2&88.8&89.3&90.6&91.7 \\\\\n$18^o$&91.3&93.5&90.9&89.5&88.2&84.6&86.2&87.1&89.8&89.4&91.8  \\\\\n$36^o$&88.5&90.4&94.1&91.7&89.1&88.4&91.0&92.5&93.8&90.1&89.4  \\\\\n$54^o$&87.5&88.3&89.7&91.8&90.3&87.6&90.6&91.1&90.4&89.2&87.1 \\\\\n$72^o$&85.6&88.5&90.4&91.6&92.4&91.3&89.1&87.3&87.0&86.3&84.4 \\\\\n$90^o$&83.0&84.7&85.3&88.6&90.8&93.7&91.7&89.6&87.4&86.2&85.1 \\\\\n$108^o$&82.4&85.6&85.4&88.1&89.5&89.4&90.0&89.8&87.2&85.3&83.2 \\\\\n$126^o$&84.6&86.1&88.6&90.3&89.3&88.0&89.8&91.9&87.5&85.7&83.4 \\\\\n$144^o$&84.7&85.0&89.2&89.8&88.2&84.4&86.9&87.8&89.9&86.5&84.5 \\\\\n$162^o$&90.5&91.2&88.8&87.2&84.6&83.4&85.1&87.4&89.7&91.5&90.2 \\\\\n$180^o$&92.0&91.6&87.8&86.8&83.5&82.4&82.9&85.1&86.5&90.9&92.4 \\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{table*}\n\nWe also conducted experiments under more challenging conditions such as a person\ncarrying an item or wearing a coat because the CASIA Dataset B includes sequences\nof such conditions, which were not used in previous studies \\cite{ref22}\\cite{ref23}\\cite{ref30}\\cite{ref31}\\cite{ref40}\\cite{ref41}. As indicated in Tables \\ref{tab:3} and \\ref{tab:4}, the $CCR$ of the gender prediction was\nsignificantly decreased under the challenging conditions of a side view or\nnearside view, even when the proper classifier was applied for the specific \nviewpoint. This problem is understandable because our viewpoint-dependent \nclassifiers are built upon sequences under normal walking conditions.\n\n \nMoreover, for a side view or nearside view, the appearance of the person is clearly changed, both when carrying an item and when wearing a coat. The mean $\\pm$ std of both $CCRs$ while carrying a bag and wearing a coat were 92.0\\% $\\pm$ 2.3 and 92.1\\% $\\pm$ 1.4, respectively. \n\nSome examples of silhouette from the same persons in front view (top row) and side view (bottom row) are shown in Fig. \\ref{fig:7}, respectively. It is interesting to notice from the figure that even in different views, the head part of the silhouette also contains the classifiable gender information (head and hair style). This is to explain that even the $90^o$ classifier is used to test the silhouette in $0^o$, the accuracy is not too low as seen in Tables 2-3-4. However, when the correct view classifier is applied, many traits for gender classification such as head and hair style, chest and back, waist and buttocks, legs  \\cite{ref23} are taken into account to increase the performance.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.5]{Fig7}\n\\caption{Silhouette example of human in front view (first row) and side view (second row)}\n\\label{fig:7}\n\\end{figure}\n\n\n\n\\begin{table*}\n\\centering\n\\caption{Results of viewpoint estimation in terms of percentage for arbitrary viewpoint under normal walking conditions }\n\\label{tab:5}      \n\\begin{tabular}{llllllllllll}\n\\hline\\noalign{\\smallskip}\n\\multirow{2}{*}{\\text{Ground-truth viewpoint}}& \\multicolumn{11}{c}{Estimated viewpoint} \\\\\n& $0^o$ & $18^o$ & $36^o$ & $54^o$ & $72^o$ & $90^o$ & $108^o$ & $126^o$ & $144^o$ & $162^o$ & $180^o$  \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n$0^o$&90.8&8.1&0&0&0&0&0&0&0&0&1.1 \\\\\n$18^o$&3.4&89.5&0&0&0&0&0&0&0&8.1&0  \\\\\n$36^o$&0&5.3&88.3&6.4&0&0&0&0&0&0&0  \\\\\n$54^o$&0&0&4.3&82.1&11.6&0&0&0&0&0&0 \\\\\n$72^o$&0&0&0&3.0&93.7&3.3&0&0&0&0&0 \\\\\n$90^o$&0&0&0&0&4.4&93.2&2.4&0&0&0&0 \\\\\n$108^o$&0&0&0&0&0&7.4&87.1&5.5&0&0&0 \\\\\n$126^o$&0&0&0&0&0&0&1.1&88.2&10.7&0&0 \\\\\n$144^o$&0&0&0&0&0&0&0&6.2&84.3&9.5&0 \\\\\n$162^o$&0&0&0&0&0&0&0&0&6.4&84.5&9.1 \\\\\n$180^o$&2.6&0&0&0&0&0&0&0&0&9.3&88.1 \\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{table*}\n\n\n\\begin{table*}\n\\centering\n\\caption{CCRs (\\%) of the viewpoint-dependent classifier of unknown viewing angle under different walking conditions without attachment-area removal module }\n\\label{tab:6}      \n\\begin{tabular}{llllllllllllll}\n\\hline\\noalign{\\smallskip}\n\\multirow{2}{*}{\\text{Walking condition}}& \\multicolumn{11}{c}{Classifier} \\\\\n& $0^o$ & $18^o$ & $36^o$ & $54^o$ & $72^o$ & $90^o$ & $108^o$ & $126^o$ & $144^o$ & $162^o$ & $180^o$ & \\text{Avg} & \\text{Unified}  \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n\\text{Normal walking}&98.5&99.1&98.7&97.8&99.3&99.8&98.7&98.4&98.3&98.9&99.2&98.8 &90.3\\\\\n\\text{Carrying backpack}&94.6&94.4&93.7&92.6&91.3&87.5&90.1&91.7&92.5&93.7&94.9&92.5 &85.7\\\\\n\\text{Wearing coat}&92.1&93.7&94.4&92.6&93.2&94.1&91.3&92.6&90.1&92.3&93.4&92.7 &86.1\\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{table*}\n\n\\begin{table*}\n\\centering\n\\caption{CCRs (\\%) of the viewpoint-dependent classifier of unknown viewing angle when including attachment-area removal module }\n\\label{tab:7}      \n\\begin{tabular}{lllllllllllll}\n\\hline\\noalign{\\smallskip}\n\\multirow{2}{*}{\\text{Walking condition}}& \\multicolumn{11}{c}{Classifier} \\\\\n& $0^o$ & $18^o$ & $36^o$ & $54^o$ & $72^o$ & $90^o$ & $108^o$ & $126^o$ & $144^o$ & $162^o$ & $180^o$ & \\text{Avg}  \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n\\text{Normal walking}&98.5&99.1&98.7&97.8&99.3&99.8&98.7&98.4&98.3&98.9&99.2&98.8 \\\\\n\\text{Carrying backpack}&94.8&94.6&94.5&94.8&93.5&95.1&94.4&94.3&93.4&93.9&94.9&94.4 \\\\\n\\text{Wearing coat}&93.1&93.8&94.6&93.4&93.7&94.5&93.1&93.5&92.3&92.6&93.7&93.5 \\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{table*}\n\n\n\\subsection{Viewpoint estimation test}\nViewpoint estimation is an important step in this work because it determines the DS model and classifier to be used for gender prediction. To test the accuracy of the viewpoint estimation module, we randomly selected ten sequences from a specific viewpoint under normal walking conditions from the CASIA Dataset B. This procedure was conducted as discussed in Section 3.2.2. The average percentages (for the ten sequences) of the viewpoint estimation are displayed in Table \\ref{tab:5}. As can be seen, given a sequence with a specific viewpoint from the CASIA Dataset B, the viewpoint estimated from the program did not match the given viewpoint (for the given $0^o$ degree sequence, the estimated viewpoints are $0^o$, $18^o$, and $180^o$ with probabilities of 90.8, 8.1, and 1.1, respectively). This is understandable because people change their gait features while walking. Moreover, the person's appearance from the front and rear views are similar, which results in a classification step, i.e., 90.8\\% for a $0^o$ classifier, 8.1\\% for an $18^o$  classifier, and 1.1\\% for a $180^o$ classifier were used to obtain the gender of the individual.\n\nAfter obtaining the viewing angle, the corresponding classifier was selected to conduct the gender prediction. Table \\ref{tab:6} displays the $CCRs$ for an unknown viewpoint under different walking conditions, i.e., normal walking, carrying an item, and wearing a coat. The CCRs under normal walking conditions (Table \\ref{tab:6}, r ow 1) are improved because the viewpoints were automatically calculated and the proper classifier was selected for the gender prediction. \n\n\nFor the given unknown $0^o$ viewpoint, $90.8\\%$ of the image sequences are estimated at $0^o$ viewing angle, $8.1\\%$ at $18^o$, and $1.1\\%$ at $180^o$, respectively, as shown in Table \\ref{tab:5}. Those image sequences are then sent to $0^o$, $18^o$, $180^o$ classifiers, respectively to calculate the $CCR$ at $0^o$ viewpoint. The performance using the corresponding classifiers under normal walking conditions increases the $CCRs$ from ($97.6\\% \\pm 0.881$) to ($98.8\\% \\pm 0.550$). The $CCRs$, while carrying a bag or wearing a coat (Table \\ref{tab:6}, rows 2 and 3), were not significantly improved because we used classifiers trained under normal walking conditions to predict the gender.\n\n\n\nFor the aim of proving the superiority of the view-dependent design, a unified classifier of all viewpoints is trained without considering the viewing angle using the same configuration of the SVM. The experimental result is reported in the last column of Table \\ref{tab:6}. As seen in the table, the average performance of view-dependent (penultimate column) increases significantly in each experimental scenario comparing to the unified classifier.\n\n\n\\begin{table*}\n\\centering\n\\caption{Comparison with related methods}\n\\label{tab:8}      \n\\begin{tabular}{llllll}\n\\hline\\noalign{\\smallskip}\n\\text{Compared methods} & \\text{\\#subjects} & \\text{Viewpoints} & \\text{Walking condition} & \\text{Reported} & \\text{Proposed method} \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n\n\\text{Lee et al. \\cite{ref42}} & \\text{14 males, 10 females} & $90^o$ & \\text{Normal} & $84.5\\%$ & $100\\%$ \\\\\n\n\\text{Li et al. \\cite{ref44}} & \\text{31 males, 31 females} & $90^o$ & \\text{Normal} & $93.2\\%$ & $99.8\\%$\\\\\n\n\\text{Yu et al. \\cite{ref23}} & \\text{31 males, 31 females} & $90^o$ & \\text{Normal} & $95.9\\%$ & $99.8\\%$\\\\\n\n\\text{Huang et al. \\cite{ref43}} & \\text{30 males, 30 females} & $0^o, 90^o, 180^o$  & \\text{Normal} & $89.5\\%$ & $99.2\\%$\\\\\n\n\\text{Zhang De \\cite{ref32}} & \\text{31 males, 31 females} & $0^o, 18^o,..., 180^o$  & \\text{Normal} & $98.1\\%$ & $98.8\\%$\\\\\n\n\\text{NA}& \\text{31 males, 31 females} & $0^o, 18^o,..., 180^o$  & \\text{Bag-carrying} & \\text{NA} & $94.4\\%$\\\\\n\n\\text{NA}& \\text{31 males, 31 females} & $0^o, 18^o,..., 180^o$  & \\text{Coat-wearing} & \\text{NA} & $93.5\\%$\\\\\n\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{table*}\n\n\\subsection{Attachment removal test}\nThe attachment area and noise can be removed using the procedure discussed in \nSection 3.3.2. During the testing phase, the silhouette was corrected and updated\nusing the attachment-area removal module. The updated version of the AGI was \ncalculated based on the new version of the silhouette. As indicated in Table \\ref{tab:6} and \nTable \\ref{tab:7}, the $CCRs$ of a human carrying a backpack in the side view ($90^o$) is \nsignificantly improved with the attachment removal module (95.1\\%), without the \nattachment removal module (87.5\\%). More improvement in the $CCRs$ of 94.4\\% $\\pm$\n0.564 and 93.5\\% $\\pm$ 0.704, compared to the cases of no attachment removal, is indicated in Table \\ref{tab:7} because of the attachment-area removal module. Moreover, a significant improvement for the side view or nearside view is presented in Table \\ref{tab:7}, row 2 (carrying a bag). The $CCR$ values in Table \\ref{tab:7}, row 1 were not changed because in this case, the person was walking with a thin coat and not carrying any objects. The attachment-area removal module did not remove anything in this case since the distance signal was within the range of $MaDS$ and $MiDS$.\n\\subsection{Comparisons}\nThe dataset used in \\cite{ref42} consists of twenty-four subjects, 14 males and 10 females, walking in normal speed and stride. The camera was placed perpendicular to their walking path. In \\cite{ref23}\\cite{ref44}experiments, only side-view sequences of 31 males and 31 females were collected from CASIA Dataset B for gender classification evaluation. Huang et al. in \\cite{ref43} extracted only 30 males and 30 females from CASIA Dataset B in three viewing angles including $0^o$,$90^o$, and $180^o$.\nTable \\ref{tab:8} presents a $CCR$ comparison of the proposed method with other related works. In the side-view dataset, the proposed method attained $CCRs$ of 100\\% and 99.8\\% compared with 84.5\\% reported in \\cite{ref42} and 95.9\\% reported in \\cite{ref23} on a small dataset and the CASIA Dataset B under the same conditions, respectively. The proposed method was also tested on normal walking conditions in three viewing angles ($0^o,90^o$, and $180^o$) and achieved greater accuracy (99.2\\% on average) compared with 89.5\\% as in \\cite{ref43}.\n\nTo demonstrate the effectiveness of the proposed method for gender classification with multiple-viewing angles, we conducted a test on multiple-views of the CASIA Dataset B (11 viewing angles) and obtained CCRs of 98.8\\%, which is also greater than the state-of-the-art method, 98.1\\% reported in \\cite{ref32}.\n\nAs described in Section 4.1, this CASIA Dataset B included three categories. The first category contained videos of humans walking in a normal condition without any attachments. The two remaining categories were more challenging, containing videos of humans carrying a backpack and humans wearing a coat. Because of the attachments, the silhouettes were highly deformed, leading to significant degradation on the classification results (Table \\ref{tab:6} and Table \\ref{tab:7}). To the best of our knowledge, there are no experimental results reported for these two remaining datasets.\n\nApplying the proposed module to remove the attachments, we performed experiments on these two datasets in multiple viewpoints (11 viewing angles) in the same scenario as the first category. The CCRs on the challenging dataset images indicated promising results of 94.4\\% and 93.5\\% for the bag-carrying and coat-wearing images, respectively, as indicated in Table \\ref{tab:8}.\n\nFurther, because the proposed method uses simple operations for gender classification such as 2-dimensional signals (distance signal), and linear SVM it requires only 48 ms (20.8 frames per second) to process a frame after skipping the first 15 frames for the AGI calculation. This means that the algorithm can be applied to a surveillance application in real time.\n\n\n\n\\section{Conclusions and future works}\nGender information can be effectively obtained from a video surveillance system based on the gait feature of the subject. Instead of using a GEI, this paper employed an AGI, which is easier to calculate for a real application. To accurately predict the human gender in real applications, we created viewpoint-dependent classifiers, i.e., a VP model and a DS model. The VP model is used to estimate the viewing angle during the testing phase; any attachment area is then removed using the DS model. Finally, the gender information is provided through the use of the viewpoint-dependent classifier. A comparison with other state-of-the-art methods \\cite{ref23} confirmed that the proposed method achieved a high accuracy of 98.8\\% and can be applied to a real-world system. However, the results of this method depend mainly on the quality of the silhouette obtained during the background subtraction step, as shown in Fig. \\ref{fig:8}. In the figure, the first row shows the samples that the method correctly classifies the gender whereas the second row shows the ones that are wrongly classified due to bad quality. As future work, we will attempt to apply the raw RGB image of a person rather than a silhouette image using deep-learning techniques because color information is an important factor for gender classification accurately predict.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[scale=0.5]{Fig8}\n\\caption{Good and bad silhouettes obtained from the background subtraction process.}\n\\label{fig:8}      \n\\end{figure}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\n{\\bf Introduction.}\nThe most general form of the neutrino electromagnetic vertex function \\cite{Giunti:2014ixa} is given by\n$\\Lambda_{\\mu}^{ij}(q) =  \\left( \\gamma_{\\mu} - {q}_{\\mu}\n\\slashed{q}\/q^{2} \\right) \\left[ f_{Q}^{ij}(q^{2}) + f_{A}^{ij}(q^{2})\nq^{2} \\gamma_{5} \\right] \\nonumber\n - i \\sigma_{\\mu\\nu} q^{\\nu} \\left[ f_{M}^{ij}(q^{2}) +\ni f_{E}^{ij}(q^{2}) \\gamma_{5} \\right]$ ,\nwhere $\\Lambda_{\\mu}(q)$ and form factors $f_{Q,A,M,E}(q^2)$ are $3\\times 3$ matrices in  the space of massive neutrinos.  In the case of coupling with a real photon ($q^2=0$) form factors provide four sets of neutrino electromagnetic characteristics: 1) the dipole magnetic moments $\\mu_{ij}=f_{M}^{ij}(0)$,\n2) the dipole electric moments $\\epsilon_{ij}=f_{E}^{ij}(0)$, 3) the millicharges $q_{ij}=f_{Q}^{ij}(0)$ and\n4) the anapole moments $a_{ij}=f_{A}^{ij}(0)$.\n\n\n\n\n{\\bf Neutrino dipole magnetic moments.} The most well understood and studied among neutrino electromagnetic characteristics are the neutrino magnetic moments.\nIn the Standard Model with massless neutrinos magnetic moments  of neutrinos are zero. Therefore, it is believed that the studies of neutrino electromagnetic properties open a window to {\\it new physics} \\cite{Giunti:2014ixa,Aprile:2020tmw,Studenikin:2008bd,Studenikin:2018vnp}. In a minimal extension of the Standard Model  the diagonal magnetic  moment of a Dirac neutrino is given \\cite{Fujikawa:1980yx} by\n$\\mu^{D}_{ii}\n  = \\frac{3e G_F m_{i}}{8\\sqrt {2} \\pi ^2}\\approx 3.2\\times 10^{-19}\n  \\Big(\\frac{m_i}{1 \\ \\mathrm{eV} }\\Big) \\mu_{B}$,\n$\\mu_B$ is the Bohr magneton. The Majorana neutrinos can have  only transition\n(off-diagonal) magnetic\nmoments  $\\mu^{M}_{i\\neq j}$. The same is valid also for the flavour neutrinos in the case of the Majorana mass states.\n\nThe most stringent constraints on the neutrino\nmagnetic moments are obtained with the reactor antineutrinos\n(GEMMA Collaboration \\cite{GEMMA:2012}):\n$\\mu_{\\nu_e} < 2.9 \\times 10^{-11} \\mu_{B}$,\nand solar neutrinos (Borexino Collaboration \\cite{Borexino:2017fbd}):\n${\\mu}_\\nu ^{eff}< 2.8 \\times\n10^{-11} \\mu _B$. The last limit can be translated to the upper limits for flavour neutrinos: $(\\mu_{\\nu_e}, \\mu_{\\nu_{\\mu, \\tau}}) \\sim (4, 6 ) \\times 10^{-11} \\mu _B$.\n\nNote that in general in the scattering experiments the neutrino\nis created at some distance from the detector as a flavor neutrino, which is a\nsuperposition of massive neutrinos. Therefore, the magnetic\nand electric moments that are measured in these experiments are not that of a\nmassive neutrino, but there are effective moments that take into account the neutrino mixing and oscillations\nduring the propagation between the\nsource and detector \\cite{Grimus:1997aa, Beacom:1999wx}.\nFor the recent and detailed study of the neutrino electromagnetic characteristics\ndependence on neutrino mixing see \\cite{Kouzakov:2017hbc}.\n\nA new phase of the GEMMA project for measuring the neutrino magnetic moment is now underway at the Kalinin Power Plant in Russia. The discussed next experiment \\cite{Belov:2015ufh} called GEMMA-3$ \/ \\nu  $GEN  is aimed at the further increase in sensitivity to the neutrino magnetic moment and will reach the level of\n$\\mu_{\\nu_e} \\sim (5{-}9) \\times 10^{-12}\\mu _B $.\nTo reach the  claimed limit on the neutrino magnetic moment  the $\\nu$GEN experiment setup reasonably improves characteristics in respect to those of the previous editions of the GEMMA project. The most important are the following \\cite{Lubashevskiy:2020}: 1) a factor of 2 increase in the total neutrino flux at the detector because of much closer location of the detector to the reactor core, 2) a factor of 3.7 increase in the total mass of the detector, 3) the energy threshold would be improved from $2.8 \\ keV$  to $ 200 \\ eV$. Furthermore, the $\\nu$GEN experimental setup is located in the new room at the Kalinin Power Plat with much better (by an order of magnitude) gamma-background conditions and on a moveable platform. The later gives an opportunity to vary online the neutrino flux and thus suppress systematic errors.\n\n\n\n\nThe observation of coherent elastic neutrino-nucleus scattering reported for the first time \\cite{Akimov:2017ade} by the\nCOHERENT experiment at the Spallation Neutron Source can be also used for constraining neutrino electromagnetic properties. For the case of neutrino magnetic moments, however, as it was shown in \\cite{Kosmas:2017tsq}\nand then confirmed in recent studies (see, for instance, \\cite{Miranda:2020tif}  ) the bounds for the flavour neutrino magnetic moments are of the order $\\mu_e , \\mu_\\mu \\sim 10^{-8} \\mu _{B}$.\n\n\n\n\nIn the recent studies \\cite{Miranda:2020kwy} it is shown that the puzzling results of the XENON1T collaboration \\cite{Aprile:2020tmw} at few keV electronic recoils  could be due to the scattering of solar neutrinos endowed with finite Majorana transition magnetic moments of the strengths lie within the limits set by the Borexino experiment  with solar neutrinos \\cite{Borexino:2017fbd}. The comprehensive analysis of the existing and new extended mechanisms for enhancing neutrino transition magnetic moments to the level appropriate for the interpretation of the XENON1T data  and leaving neutrino masses within acceptable values is provided in \\cite{Babu:2020ivd}.\n\nIn the most recent paper \\cite{Cadeddu:2019qmv} we have proposed an experimental setup to observe coherent elastic neutrino-atom scattering using electron antineutrinos from tritium decay and a liquid helium target. In this scattering process with the whole atom, that has not beeen observed so far, the electrons tend to screen the weak charge of the nucleus as seen by the electron antineutrino probe.\n Finally, we study the sensitivity of this apparatus to a possible electron\n neutrino magnetic moment and we find that it is possible\n to set an upper limit of about\n$\\mu_{\\nu} < 7 \\times 10^{-13} \\mu_{B}$,\nthat is more than one order of magnitude smaller than\nthe current experimental limits from GEMMA and Borexino.\n\n\n\nAn astrophysical bound on an effective neutrino magnetic moment (valid for both cases of\nDirac and Majorana neutrinos) is provided\n\\cite{Raffelt-Clusters:90, Viaux-clusterM5:2013, Arceo-Diaz-clust-omega:2015}\nby observations of the properties of globular cluster stars:\n$\\Big( \\sum _{i,j}\\left| \\mu_{ij}\\right| ^2\\Big) ^{1\/2}\\leq (2.2{-}2.6) \\times\n10^{-12} \\mu _B$. There is also a statement \\cite{deGouvea:2012hg}, that\nobservations of supernova fluxes  in the future largevoluem experiments like JUNO, DUNE and Hyper-Kamiokande ( see for instance\n\\cite{An:2015jdp,Giunti:2015gga,Lu:2016ipr}) may reveal the effect of  collective  spin-flavour oscillations  due to the Majorana neutrino transition moment $\\mu^{M}_\\nu \\sim 10^{-21}\\mu_B$. Other new possibilities for neutrino\nmagnetic moment visualization in extreme astrophysical environments are\nconsidered recently in \\cite{Grigoriev:2017wff,Kurashvili:2017zab}.\n\n\n\nA general and termed model-independent upper bound on the Dirac neutrino\nmagnetic moment, that can be generated by an effective theory beyond\na minimal extension of the Standard Model, has been derived in\n\\cite{Bell:2005kz}: $\\mu_{\\nu}\\leq\n10^{-14}\\mu_B$. The corresponding limit for transition moments of Majorana neutrinos is much weaker \\cite{Bell:2006wi}.\n\n\n{\\bf Neutrino dipole electric moments.} In the theoretical framework with $CP$ violation a neutrino\ncan have nonzero electric moments $\\epsilon_{ij}$. In the laboratory neutrino\nscattering experiments for searching $\\mu_{\\nu}$ (for instance, in the GEMMA experiment)\nthe electric moment $\\epsilon_{ij}$ contributions interfere with\nthose due to $\\mu_{ij}$. Thus, these kind of experiments also provide constraints\non $\\epsilon_{ij}$. The astrophysical bounds on $\\mu_{ij}$\nare also applicable for constraining $\\epsilon_{ij}$ (see \\cite{Raffelt-Clusters:90, Viaux-clusterM5:2013, Arceo-Diaz-clust-omega:2015} and \\cite{Raffelt:2000kp}).\n\n{\\bf Neutrino electric millicharge.} There are extensions of the Standard Model that allow for nonzero\nneutrino electric millicharges. This option can be provided by\nnot excluded experimentally possibilities for hypercharhge dequantization or\nanother {\\it new physics} related with an additional $U(1)$ symmetry\npeculiar for extended theoretical frameworks. Note that neutrino millicharges\nare strongly constrained on the level $q_{\\nu}\\sim 10^{-21} e_0$\n($e_0$ is the value of an electron charge) from neutrality of the hydrogen atom.\n\n A nonzero neutrino millicharge $q_{\\nu}$ would contribute to the neutrino electron scattering in the terrestrial experiments. Therefore, it is possible to get bounds on $q_{\\nu}$ in the reactor antineutrino\n experiments. The most stringent reactor antineutrino constraint\n\n $q_{\\nu}< 1.5 \\times 10^{-12} e_0 $\n\n is obtained in \\cite{Studenikin:2013my} within the free-electron approximation using the GEMMA experimental data \\cite{GEMMA:2012}. This limit is cited by the Particle Data Group since 2016 (see also \\cite{Zyla:2020zbs}).\nA certain increase in the cross section is expected in the case when instead of the free-electron approximation one accounts for\nthe so called atomic ionization effect \\cite{Chen:2014dsa}, and the obtained corresponding limit on the neutrino millicharge is\n $q_{\\nu} < 1 \\times 10^{-12} e_0$.\n\n\n The expected increasing sensitivity to the neutrino-electron scattering of the future $\\nu$GEN experiment that is aimed to reach a new  limit  for the magnetic moment would provide a possibility \\cite{Studenikin:2013my} to check the neutrino millicharge at the scale of  $ q_{\\nu} \\sim  10^{-13} e_0$.\n\nAs it has been already mentioned above, the coherent elastic neutrino-nucleus scattering \\cite{Akimov:2017ade} is a new\npowerful tool to probe the electromagnetic neutrino properties \\cite{Kosmas:2017tsq}. In the flavour basis neutrinos can have diagonal $q_{lf}$ ($l=f$, $l,f = e, \\mu, \\tau$) and transition $q_{lf}$ ($l\\neq f $) electric charges (see, for instance, \\cite{Giunti:2014ixa} and \\cite{Kouzakov:2017hbc}). Such possibilities are not excluded by theories beyond the Standard Model. Recently \\cite{Cadeddu:2019eta} from the analysis of the COHERENT data new constraints for all neutrino charges on the level of $\\sim 10^{-7} e_0$ are obtained. It follows, that the bounds for involving the electron neutrino flavour charges $q_{ee}, q_{e\\mu}$ and $q_{e\\tau}$ are not competitive with respect to constraints $\\sim 10^{-12} e_0$ obtained for the effective electron neutrino charge $q_{eff}= \\sqrt{q_{ee}^2 +q_{e\\mu}^2 +q_{e\\tau}^2}$ from the reactor antineutrino scattering experiments \\cite{Studenikin:2013my, Chen:2014dsa}. Note, that the bounds for $q_{\\mu \\mu}$ and $q_{\\mu \\tau}$ from a laboratory data are obtained in \\cite{Cadeddu:2019eta} for the first time.\n\n\nThe most recent and one of the most detailed statistical studies \\cite{Parada:2019gvy} of experimental data from the elastic neutrino-electron and coherent neutrino-nucleus scattering show that the combined inclusion of different experimental data can lead to stronger\nconstraints on $q_\\nu$ than those based on individual analysis of different experiments.\n\n\nA neutrino millicharge would have specific phenomenological consequences\nin astrophysics because of new electromagnetic processes are opened\ndue to a nonzero charge (see \\cite{Giunti:2014ixa,Raffelt:1996wa,Studenikin:2012vi}). Following this line, the most stringent astrophysical constraint on neutrino millicharges\n$q_{\\nu}< 1.3 \\times 10^{-19} e_0 $\n was obtained in \\cite{Studenikin:2012vi}. This bound\nfollows from the impact of the {\\it neutrino star turning} mechanism ($\\nu ST$) \\cite{Studenikin:2012vi} that can be considered as a {\\it new physics} phenomenon end up with a pulsar rotation frequency\nshift engendered by the motion of escaping from the\nstar neutrinos along curved trajectories due to millicharge interaction with a constant\nmagnetic field of the star. The existed other astrophysical constraints on the neutrino millicharge, however less restrictive than that of \\cite{Studenikin:2012vi}, are discussed in \\cite{Giunti:2014ixa,Parada:2019gvy}.\n\n{\\bf Neutrino cherge radius and anapole moment.} Even if a neutrino millicharge is vanishing, the electric form factor\n$f^{ij}_{Q}(q^{2})$ can still contain nontrivial information about\nneutrino electromagnetic properties. The corresponding electromagnetic characteristics is\ndetermined by the derivative of $f^{ij}_{Q}(q^{2})$ over $q^{2}$  at\n$q^{2}=0$ and is termed neutrino charge radius,\n$\\langle{r}_{ij}^{2}\\rangle\n=-\n6\n\\frac{df^{ij}_{Q}(q^{2})}{dq^{2}} \\\n_{\\mid _ {q^{2}=0}}\n$ (see \\cite{Giunti:2014ixa} for the detailed discussions).\nNote that for a massless neutrino the neutrino charge radius is the only\nelectromagnetic characteristic that can have nonzero value. In the Standard Model\nthe neutrino charge radius and the anapole moment are not defined separately,\nand there is a relation between these two values: $a = - \\frac{\\langle{r}^{2}\\rangle}{6}$.\n\nA neutrino charge radius contributes to the neutrino scattering cross section on electrons and thus\ncan be constrained by the corresponding laboratory experiments \\cite{Bernabeu:2004jr}.\nIn all but one previous studies it was claimed\n that the effect of the neutrino\ncharge radius can be included just as a shift of the vector coupling constant $g_V$\nin the weak\ncontribution to the cross section.\nHowever, as it has been recently demonstrated in \\cite{Kouzakov:2017hbc} within the direct calculations of\nthe elastic neutrino-electron scattering cross section accounting for all possible neutrino electromagnetic characteristics\nand neutrino mixing, this is not the fact. The neutrino charge radius dependence of the cross section\nis more complicated and there are, in particular, the dependence on the interference terms of the type\n$g_{V}\\langle{r}_{ij}^{2}\\rangle$ and also on the neutrino mixing.\nThe current constraints on the flavour neutrino charge radius $\\langle{r}_{e,\\mu,\\tau}^{2}\\rangle\\leq 10^{-32} - 10^{-31} \\ cm ^2$\nfrom the scattering experiments differ only by 1 to 2\norders of magnitude from the values $\\langle{r}_{e,\\mu,\\tau}^{2}\\rangle\\leq 10^{-33} \\ cm ^2$ calculated within the minimally extended Standard Model with right-handed neutrinos\n\\cite{Bernabeu:2004jr}. This indicates that the minimally extended Standard Model neutrino charge radii could be experimentally tested in the near future.\n\nNote that there is a need to re-estimate experimental constraints on\n$\\langle{r}_{e,\\mu,\\tau}^{2}\\rangle$  from the scattering experiments following\nnew derivation of the cross section \\cite{Kouzakov:2017hbc} that properly accounts for the interference of the weak and charge radius electromagnetic interactions and also for the neutrino mixing.\n\nRecently constraints on  charged radii  have been obtained\n\\cite{Caddedu:2018prd} from the analysis of the data on coherent\nelastic neutrino-nucleus scattering obtained in the COHERENT experiment\n\\cite{Akimov:2017ade,Akimov:2018vzs}. In addition to the customary diagonal\ncharge radii $\\langle{r}_{e,\\mu,\\tau}^{2}\\rangle$\nalso the neutrino transition (off-diagonal) charge radii have been constrained\nin \\cite{Caddedu:2018prd} for the first time:\n$\\left(|\\langle r_{\\nu_{e\\mu}}^2\\rangle|,|\\langle r_{\\nu_{e\\tau}}^2\\rangle|,|\\langle r_{\\nu_{\\mu\\tau}}^2\\rangle|\\right)\n< (22,38,27)\\times10^{-32}~{\\rm cm}^2$. Since 2018 these limits are included by the Particle Data Group to Review of Particle Properties (see also \\cite{Zyla:2020zbs}) and also were noted by the Editors' Suggestion as the most important results (PRD Highlights 2018) published in the journal.\n\n\n\nThe work is supported by the Russian Foundation for Basic Research under grant No. 20-52-53022-GFEN-a.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nResearch on highly accurate atomic frequency standards and their applications is making fast and steady progress. The quest for ever increased accuracy is advancing hand in hand with advances in quantum physics, with better understanding and manipulation of atomic systems, with exploration of fundamental laws of nature and with the development of important services and infrastructures for science and society. The quest for increased accuracy is also a powerful incentive to innovation in such areas as lasers, laser stabilization, low noise electronics, stable oscillators, low noise detection of optical signals, fiber devices and cold-atom based instrumentation for ground or space applications. Finally, in addition to enhancing existing applications, improved accuracy leads to new applications.\n\nThis article focuses on key achievements and trends of the last 10 years. Over this period of time, the first generation of laser-cooled standards, using the atomic fountain geometry, reached maturity and had large impact on international timekeeping. The typical accuracy of these frequency standards,  2 parts in $10^{16}$, is now permanently accessible both locally and globally. At the same time, a new generation of optical clocks showed tremendous and steady improvement, gaining more than two orders of magnitude in a decade. To date, an accuracy of $6.4\\times 10^{-18}$ \\cite{bloom2014optical} was reported. Similarly, major improvements occurred in many other aspects of optical frequency metrology, notably in optical frequency combs and optical fiber links.\n\nIn this article, we will report on developments of the LNE-SYRTE atomic clock ensemble since our 2004 report in the Special Issue of the Comptes Rendus de l'Acad\\'emie des sciences on Fundamental Metrology \\cite{Bize2004}. This work exemplifies many of the above mentioned features of research on highly accurate atomic frequency standards. We will focus on frequency standards and their impact on timescales and timekeeping, on clock comparisons, including optical-to-microwave comparisons with combs, and their applications. Several other aspects of our research are covered by other articles of the Special Issue of the Comptes Rendus de l'Acad\\'emie des Sciences (Volume 16, Issue 5), i.e. the space mission PHARAO\/ACES \\cite{cras2014onACES}, fundamental tests with clocks \\cite{cras2014onFundamentalTests}, development of technologies for space optical clocks \\cite{cras2014onSpaceOpticalClocks} and optical fibers links \\cite{cras2014onFiberLinks1,cras2014onFiberLinks2}.\n\n\n\\section{Atomic fountains} \\label{fountains}\n\n\\begin{figure*}[t]\n\t\\centering\n  \\includegraphics[width=1\\textwidth, angle=0]{Fig1_ClockEnsemble1.pdf}\n\t\\caption{Overview of the LNE-SYRTE atomic clock ensemble at the Observatoire de Paris.}\n\t\\label{fig:ClockEnsemble}\n\\end{figure*}\n\nAtomic fountains are the first generation of the laser-cooled atomic frequency standards. They use the fountain geometry where spectroscopy of the clock transition is performed onto a free-falling sample of laser-cooled atoms which is beforehand launched upwards vertically (see, for instance, \\cite{guena2012} and references therein). To date, atomic fountains using cesium provide the most accurate realization of the SI second. One important aspect of the last 10 years was to better understand systematic shifts limiting the accuracy of these devices.\n\n\\textit{Distributed cavity phase shift}-- In atomic fountains, the Ramsey interrogation is realized by the up-going and down-going passages of the atomic sample through the microwave Ramsey cavity. The spatial phase variations of the field inside the cavity, when sampled by the moving atoms, induce a frequency shift, which can be described as a residual Doppler shift. For a long time, there was a lack of both a complete and agreed model for this effect and of experiments to test it. Consequently, this effect was one of the main sources of uncertainty in atomic fountains. In \\cite{li2004,li2010}, a new approach was proposed to compute the cavity phase distribution. We performed measurements of these shifts in FO2-Cs which enabled the first quantitative comparison between theory and experiment \\cite{guena2011}. This study validated the theoretical model and lowered the distributed cavity phase uncertainty for FO2-Cs to $10^{-16}$. It also defined a method to determine this uncertainty, which was then adopted in \\cite{li2011,weyers2012} and for other SYRTE fountains.\n\n\\textit{Microwave lensing shift}-- The microwave field inside the Ramsey cavity not only excites the transition between the two internal clock states but also modifies the motion of atomic wave packets, leading to a frequency shift \\cite{Borde2002}\\cite{Wolf2004}. In \\cite{Gibble2006}, a new approach to compute the shift was proposed. It was then used in a complete model of the effect, taking into account all features of the interaction such as atomic velocity and space distributions and detection non-uniformities \\cite{li2011}\\cite{weyers2012}. This same method was applied to LNE-SYRTE fountains, for which shifts are reported in table 2 of \\cite{guena2012}.\n\n\\textit{Blackbody radiation shift}-- In 2004, conflicting measurements and calculations of this shift induced by thermal radiation bathing the atoms were reported. This led us to revisit our early accurate measurements of the Stark coefficient \\cite{simon1998}. Our new measurements at lower electric fields have been found to be in excellent agreement \\cite{rosenbusch2007}. The theory of the Stark shift developed in the 60's turned out to have a sign error for the tensor part. This led to a small change of the blackbody radiation shift correction of $7\\times 10^{-17}$ \\cite{guena2012}. Two independent high accuracy ab initio calculations further agreed with the blackbody radiation shift correction derived from our Stark measurements.\n\n\\textit{Microwave leakage and synchronous phase perturbations}-- Interaction of atoms with unintended residual microwave field and synchronous perturbation of phase of the probing field can produce shifts. We developed microwave synthesizers that can be switched without introducing phase transients and a phase transient analyzer with 1~$\\mu$rad.s$^{-1}$ resolution \\cite{santarelli2009}. Using these tools, we lowered the uncertainty related to these putative frequency shifts to less than $10^{-16}$.\n\nOur approach to deal with other systematic shifts remained as described in our last report in the Comptes Rendus \\cite{Bize2004}. Table \\ref{tab_accuracy} gives, as an example, the accuracy budget of FO2-Cs as of 2014.\n\n\\begin{table}\n\\footnotesize\n\\caption{Typical uncertainty budget of the FO2-Cs primary frequency standard (top). Uncertainty budget of the Sr1 optical lattice clock as of July 2011 (bottom). Tables give the fractional frequency correction and its Type~B uncertainty for each systematic shift, in units of $10^{-16}$. The total uncertainty is the quadratic sum of all uncertainties.}\\label{tab_accuracy}\n\\begin{center}\n\\begin{tabular}{lcc}\n\\multicolumn{3}{c}{FO2-Cs}\\\\\n\\hlin\nPhysical origin of the shift                                       & Correction   & Uncertainty  \\\\\n\\hline\n{\\small Quadratic Zeeman}               & $-1919.9$&$0.3$  \\\\\n{\\small Blackbody radiation}                  &   $168.4$&$0.6$\\\\\n{\\small Collisions and cavity pulling}        &   $201.2$&$1.5$\\\\\n{\\small Distributed cavity phase}       & $-0.9$& $1.2$   \\\\\n{\\small Microwave lensing}                     &  $-0.7$&$0.7$   \\\\\n{\\small Spectral purity \\& leakage}            & 0  &$0.5$ \\\\\n{\\small Ramsey \\& Rabi pulling}                & 0 &$0.1$  \\\\\n{\\small Relativistic effects}            & 0 & $0.05$ \\\\\n{\\small Background collisions}                 & 0  &$1.0$ \\\\\n\\hline\n{\\small Total}                               &  $-1551.9$&$2.5$  \\\\\n\\hline\n\\end{tabular}\n\\hspace{1cm}\n\\begin{tabular}{lcc}\n\\multicolumn{3}{c}{Sr1}\\\\\n\\hlin\nPhysical origin of the shift                                       & Correction   & Uncertainty  \\\\\n\\hline\n{\\small Quadratic Zeeman}               & $19.7$&$0.2$  \\\\\n{\\small Blackbody radiation}                  &   $ 53.8 $&$0.8$\\\\\n{\\small Collisions}        &   $-0.2 $&$0.5 $\\\\\n{\\small AC Stark shift lattice 1st order}       & $-0.5 $& $0.1 $   \\\\\n{\\small AC Stark shift Lattice 2nd order}      &  $0 $&$ 0.1$   \\\\\n{\\small DC Stark shift}            &  0  &$0.01 $ \\\\\n{\\small Line pulling}                & 0 &$0.5$  \\\\\n\\hline\n{\\small Total}                               &  $72.8$&$1.0$  \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\vspace{-6mm}\n\\end{table}\n\n\\normalsize\n\\vspace{1mm}\n\nAnother important achievement was the simultaneous operation with $^{87}$Rb and $^{133}$Cs of the dual fountain FO2. This was done by implementing dichroic collimators overlapping 780~nm (for Rb) and 852~nm (for Cs) radiations for all laser beams, and by adopting a time sequence enabling time resolved selective detection of the two atomic species \\cite{guena2010}. Further notable improvement relates to reliability and capability for long term unattended operation. Using 2D magneto-optic traps to load the optical molasses suppressed residual background vapor and enhanced the lifetime of the alkali sources. Also, we developed an automatic data processing system that monitors the status of all fountains, oscillators and internal links in quasi real time. This system allows rapid detection of failures. It also performs automated fountain data processing, taking account of all systematic corrections, and it continuously generates frequency measurements at the nominal uncertainty of the fountains. This capability has had major impact on timekeeping and other applications of atomic fountains (see sect.~\\ref{sec:fundphystests} and \\ref{sec:timekeeping} ).\n\n\\section{Optical lattice clocks}\n\n\\label{sec:OLC}\n\nIn optical lattice clocks (OLCs), a set of neutral cold atoms, dipole-trapped in an optical lattice, are interrogated by an ultra-stable ``clock'' laser. Because they involve probing an optical transition of a large (typically $10^4$) number of tightly confined atoms, they combine an excellent ultimate frequency stability -- only limited by the Quantum Projection Noise (QPN) -- and a high accuracy. Proposed in 2001 \\cite{Katori2001}\\cite{Katori2003a}, OLCs made tremendous progress in the last decade. OLCs have demonstrated unprecedented frequency stabilities of a few $10^{-16}\/\\sqrt{\\tau}$ and a record accuracy below $10^{-17}$ \\cite{bloom2014optical}, overcoming the best ion clocks~\\cite{letargat2013,hinkley_atomic_2013,falke2014,bloom2014optical,ushijima2014cryogenic}. With current improvement in laser stabilization, OLCs are expected to reach a QPN limited stability on the order of $10^{-17}\/\\sqrt{\\tau}$ within a few years, thus enabling even better characterization of systematic effects. OLCs with Sr, Yb, Hg and more prospectively Mg have been demonstrated. Among these atomic species, Sr is currently the most popular choice because of the accessibility of the required laser wavelengths, the possibility to cool Sr down to sub-$\\mu$K temperature using the narrow $^1S_0 \\rightarrow{}^3P_1$ inter-combination line, and the possibility it offers on the control of systematic effects, most notably concerning  the high order perturbation by the trapping light and cold collisions.\n\n\\textit{Strontium optical lattice clocks}-- LNE-SYRTE developed two OLCs using strontium atoms. The design of these clocks uses an optical cavity to enhance the optical lattice light, giving access to large trap depths. It enabled us to explore systematic effects induced by the trapping laser light. These effects are specific to OLCs, and we have demonstrated that they can be controlled to better than $10^{-17}$, even with a significant trapping depth~\\cite{Brusch2006a,westergaard2011lattice}, thus validating the concept of OLCs. In particular, we have determined the precise value for the ``magic wavelength'' for which the impact of the trapping light is canceled to first order, and resolved or upper-bounded a number of higher order effects. Because OLCs use a large number of atoms in a tightly confined space, they are subject to a significant density-dependent systematic frequency shift. Some groups have resolved a density shift on the order of $10^{-16}$ with both Sr and Yb. However, the loading technique chosen at LNE-SYRTE leads to a lower atomic density, thus dramatically reducing this effect below $10^{-17}$. The blackbody radiation shift has remained the dominant contribution to the accuracy budget, with an uncertainty around $5\\times 10^{-17}$ for both Sr and Yb, assuming a 1~K uncertainty on the temperature of the environment. Recently, precise measurements of the static polarizability of Yb and Sr, together with carefully crafted environments for the atoms has enabled a few groups to drastically reduce this uncertainty, down to the $10^{-18}$ range.\n\nComparisons between clocks are necessary to confirm their accuracy budget. The first comparisons between remote Sr OLCs were achieved by comparing to cesium clocks (see sect.~\\ref{sec:fundphystests}), but they were soon limited by the cesium accuracy. LNE-SYRTE published the first comparison between two local OLCs that confirm the accuracy budget of the clocks better than the accuracy of the cesium fountains, involving two Sr clocks with a frequency difference smaller than their combined accuracy budgets of $1.5\\times 10^{-16}$~\\cite{letargat2013}. This resolution is obtained after less than one hour of integration. Table~\\ref{tab_accuracy} gives the accuracy budget of one of the Sr OLCs at the time of this comparison.\n\n\n\\textit{Mercury optical lattice clock}-- SYRTE also started the development of a mercury OLC. Hg has the advantage of low sensitivity to blackbody radiation and to electric field (30 times lower than Sr, 15 times lower than Yb). For the $^{199}$Hg isotope, clock levels have a spin 1\/2 for which the tensor light shift sensitivity is absent. Also, because of its high vapor pressure, it does not require an oven and enables the use of a 2D magneto-optic trap as the initial source of atoms. Hg is also interesting for fundamental physics and atomic physics, because of a quite high sensitivity to a variation of $\\alpha$ (see sect.~\\ref{sec:fundphystests}) and its 7 natural isotopes. The main challenge of using Hg lies in the need for deep UV laser sources. When the potential of Hg for a highly accurate optical lattice clock arose, Hg had never been laser cooled.\n\nIn the last years, we made all the steps leading to the demonstration, for the first time, of a Hg lattice clock. Laser cooling on the 254 nm $^1S_ 0 \\rightarrow{} ^3P_1$ intercombination transition was demonstrated and studied \\cite{Petersen2008b}\\cite{Hachisu2008}\\cite{McFerran2010}. A clock laser system with thermal noise limited instability of $4\\times 10^{-16}$ was developed \\cite{Millo2009b}\\cite{Dawkins2010}. We performed the first direct laser spectroscopy of the the clock transition, firstly on atoms free-falling from a magneto-optic trap \\cite{Petersen2008a} and secondly on lattice-bound atoms, with linewidth down to 11~Hz (at 265.6~nm or 1128~THz). We performed the first experimental determination of the ``magic wavelength'' \\cite{Yi2011}, for which our best value is $362.5697 \\pm \ufffd0.0011$~nm. We performed initial measurements of the absolute frequency of the $^{199}$Hg clock transition down to an uncertainty of 5.7 parts in $10^{15}$ \\cite{McFerran2012}.\n\nSo far, the advancement of the Hg optical lattice was mainly hindered by the poor reliability of the 254~nm laser-cooling and by the modest lattice trap depth. Recent work enabled large improvements to overcome these two limitations, opening the way to in-depth systematic studies and higher accuracies.\n\n\n\n\n\\section{Optical frequency combs}\n{}\nThe development of optical frequency standards of the kind presented in the previous section is aimed at producing a laser electromagnetic field of extremely stable and accurate frequency. To realize a complete metrological chain that allows comparison with other standards operating at different wavelengths in the optical domain or in the microwave domain (such as primary frequency standards), it is necessary to use a specific device. The method of choice nowadays is the optical frequency comb based on mode-locked femto-second laser, that provides a phase coherent link spanning across the optical and microwave domains. Recently, SYRTE focused on a technology of comb based on erbium-doped fiber lasers, whose foremost asset is the capability to function for months with a very limited maintenance while performing state-of-the-art measurements. Tight phase locking (bandwidth $\\sim 1\\,$MHz) of such comb onto a 1.5~$\\mu$m ultra-stable laser sets it in the narrow-linewidth regime where beatnotes with other ultra-stable light at other wavelengths are typically $\\sim1\\,$Hz linewidth. This provides high performance simultaneous measurements of the various ultra-stable optical frequency references at SYRTE in the visible and near infrared domain.\n\nThe comb also behaves like a frequency divider: the repetition rate of the comb, $f_{\\rm rep}$, results from the coherent division of the 1.5$\\,\\mu$m laser frequency $\\nu_{\\rm L}$. By photo-detecting the train of pulses and filtering out one specific harmonic of the repetition rate, one can generate a microwave signal whose phase noise is that of the cw optical reference, divided by the large frequency ratio (typically 20000) between a 1.5$\\,\\mu$m wavelength cw laser and a $10\\,$GHz microwave signal. $10\\,$GHz signals with phase noise of -100\\,dBc\/Hz at 1\\,Hz Fourier frequency and -140\\,dBc\/Hz white noise plateau are now straightforward to produce. This low phase noise level is comparable with that of cryogenic sapphire oscillators and is thus sufficient to operate state-of-the-art microwave atomic fountains (sect.~\\ref{fountains}) with a short term stability limited only by atomic quantum projection noise. We have realized proof-of-principle experiments of such a scheme \\cite{MilloAPL2009} and are now progressing toward implementing it in an operational system. We further demonstrated several advanced techniques \\cite{ZhangAPB2012, HabouchaOL2011, ZhangOL2014} to reduce the imperfection of the frequency division and photo-detection processes. We have shown that it is now becoming possible to generate microwave signals with phase noise as low or lower than any other technology for a large range of Fourier frequencies. Applications of such extremely low noise microwave signals can be found in RADAR (civil and military) as well as very long baseline interferometry.\n\nFinally, the comb-based transfer of spectral purity between different wavelengths recently led to exciting results. This application is crucial for the future development of optical lattice clocks, whose short term stability is currently limited by the spectral purity of the ultra-stable clock laser probing the atomic transition. Several competing technologies are being explored, notably at SYRTE, to improve the performance of these lasers, all of them very challenging, some of them wavelength-specific. Being able to utilize the performance of an extremely stable laser at a given wavelength and to distribute its performance to any wavelength within reach of the frequency comb is an important milestone for the future developments of optical lattice clocks. We demonstrated such transfer from a master to a slave laser with an added instability of no more than a few $10^{-18}$ at $1\\,$s (see fig. \\ref{fig:TSP}, right), well within the requirements expected for the next several years \\cite{NicolodiNP2014}.\n\n\\begin{figure*}[ht]\n\t\\centering\n  \\includegraphics[width=0.9\\textwidth, angle=0]{Fig2_FrequencyComb.pdf}\n\t\\caption{Principle of the transfer of spectral purity (left): the optical beatnotes of the comb with a master laser on the one hand, and with a slave laser on the other hand, are rescaled and mixed before being compared to a stable synthesizer. The feedback on the offset-locked slave transfers the spectral purity of the master to the slave laser. The modified Allan deviation (right) of the noise added by the transfer itself has a level of only $2\\times10^{-18}$ at $1\\,$s, and averages down to $2 \\times 10^{-20}$ after $1000\\,$s.}\n\t\\label{fig:TSP}\n\\end{figure*}\n\n\\section{Fundamental physics tests}\\label{sec:fundphystests}\n\nOne exciting scientific application of atomic clocks with extreme uncertainties is to contribute to testing fundamental physical laws and searching for physics beyond the Standard Model of particle physics. The frequency of an atomic transition relates to parameters of fundamental interactions (strong interaction, electro-weak interaction), such as the fine-structure constant $\\alpha$, and to fundamental properties of particles like for instance the electron mass, $m_e$. Repeated highly accurate atomic clock comparisons can be used to look for a putative variation with time or with gravitational potential of atomic frequency ratios, and, via suitable atomic structure calculations, of natural constants. Clocks provide laboratory tests, independent of any cosmological model, that constrain alternative theories of gravity and quantum mechanics, thereby contributing to the quest for a unified theory of the three fundamental interactions.\n\n\n\\label{RbCsFitLin}\n\\textit{$^{87}$Rb vs $^{133}$Cs comparisons}-- Improvements of atomic fountains described in sect.~\\ref{fountains} enabled major enhancement in the number and in the quality of Rb\/Cs hyperfine frequency ratio measurements since our last report in the Comptes Rendus \\cite{Bize2004}. Measurements have been performed almost continuously since 2009. Fig.\\ref{Graphdrift}, top shows the temporal record of the variations of this ratio. Measurements extending over 14~years give stringent measurements of a putative variation with time and gravity\nof the Rb\/Cs ratio, as reported in \\cite{guena2012b}. Taking into account out most recent data, we get $d \\ln(\\nu_{\\mathrm{Rb}}\/\\nu_{\\mathrm{Cs}})\/dt=(-11.6\\pm 6.1 )\\times 10^{-17}$~yr$^{-1}$ for the time variation. For the variation scaled to the annual change of the Sun gravitational potential on Earth $U$, we get $c^2 d \\ln(\\nu_{\\mathrm{Rb}}\/\\nu_{\\mathrm{Cs}})\/dU=(7.4 \\pm 6.5)\\times10^{-7}$, which provides a differential redshift test between Rb and Cs twice more stringent than \\cite{peil2013}.\n\n\\begin{figure}\n\t\\begin{center}\n\n\t \\includegraphics[width=0.45\\textwidth]{Fig3_RbCs.pdf}\n\n\t \\vspace{3mm}\n\n \\includegraphics[width=0.48\\textwidth]{Fig3_SrCs.pdf}\n \n\t\\end{center}\n\n\t \\vspace{1mm}\n\n\\footnotesize\n\\begin{center}\n\\begin{tabular}{p{28mm} ccc}\n\\hline \\hline\n & $\\ln(\\alpha)$ &$\\ln(\\mu$)&$\\ln(m_{q}\/\\Lambda_{\\mathrm{QCD}})$ \\\\\n\\hline\n$d\/dt~ (\\times 10^{-16}\\mathrm{yr}^{-1})$& $-0.26\\pm 0.24$ &$1.1\\pm 1.4$& $59\\pm 30$\\\\\n$c^{2}d\/dU ~(\\times 10^{-6})$ & $0.27 \\pm 0.46$ & $-0.2 \\pm 2.1$ & $-2.9\\pm 5.6$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\normalsize\n\n\\caption{Top: Temporal record of fractional variations of the $\\nu_{\\mathrm{Rb}}\/\\nu_{\\mathrm{Cs}}$ hyperfine frequency ratio. The error bars are the total 1~$\\sigma$ uncertainties, dominated by the systematic uncertainties. The solid red line is the weighted fit to a line with inverse quadratic weighting. The origin of the vertical axis corresponds to the $^{87}$Rb secondary representation of the SI second recommended in 2012, with a recommended uncertainty $1.2 \\times 10^{-15}$ (grey area) \\cite{CCTF2012}. Middle: International comparisons of the $\\nu_{\\mathrm{Sr}}\/\\nu_{\\mathrm{Cs}}$ frequency ratio. A fit shows an upper bound on a drift of this ratio, as well as on a variation synchronized with the Earth's orbit around the Sun. Bottom: Results of the global analysis of accurate experimental determinations of variations of atomic frequency ratios, available as of October 2014. The table gives constraints on temporal variations and on couplings to gravitational potential for the three fundamental constants: $\\alpha$, $\\mu=m_{e}\/m_{p}$ and $m_{q}\/\\Lambda_{\\mathrm{QCD}}$.} \\label{Graphdrift}\n\\end{figure}\n\n\n\\textit{$^{87}$Sr vs $^{133}$Cs comparisons}-- Frequency ratios between optical and microwave clocks offer a different sensitivity to natural constants than hyperfine frequency ratios. International absolute frequency measurements of strontium optical lattice clocks against Cs fountain primary frequency standards over a decade (fig.\\ref{Graphdrift}, Middle) give the linear drift with time of the $\\nu_{\\mathrm{Sr}}\/\\nu_{\\mathrm{Cs}}$ ratio: $d \\ln(\\nu_{\\mathrm{Sr}}\/\\nu_{\\mathrm{Cs}})\/dt=(-2.3 \\pm 1.8)\\times 10^{-16}$~yr$^{-1}$, and of the variation with the gravitational potential: $c^2 d \\ln(\\nu_{\\mathrm{Sr}}\/\\nu_{\\mathrm{Cs}})\/dU=(-1.3 \\pm 1.5)\\times 10^{-6}$. Because the accuracy of these measurements has improved considerably over time, these bounds will be significantly improved by future measurements.\n\n\\textit{Combining with other comparisons}-- Each pair of atoms has a different sensitivity to variations of three fundamental constants $\\alpha$, $\\mu=m_e\/m_p$ and $m_{q}\/\\Lambda_{\\mathrm{QCD}}$. To set independent limits to variations of these three constants with time, one can perform weighted least-squares fit to all accurate experimental determinations of variation with time of atomic frequency ratios, available as of October 2014. This includes the above Rb\/Cs result (see sect.~\\ref{RbCsFitLin}), optical frequency measurements of H(1S-2S) \\cite{fischer2004}, Yb$^{+}$ \\cite{tamm2014}, Hg$^{+}$ \\cite{fortier2007}, Dy \\cite{leefer2013} and Sr (see above, and \\cite{letargat2013} and references therein) against Cs, and the optical-to-optical ion clock frequency ratio $\\mathrm{Al}^{+}\/\\mathrm{Hg}^{+}$ \\cite{rosenband2008}.\nThe fit yields independent constraints for the three constants given in the first row of the Table in fig.~\\ref{Graphdrift}. The constraint relative to $\\alpha$ is mainly determined by the $\\mathrm{Al}^{+}\/\\mathrm{Hg}^{+}$ comparison. In this fit, only the Rb\/Cs comparison disentangles variations of $\\mu$ and of $m_q\/\\Lambda_{\\mathrm{QCD}}$. It is therefore essential to constrain $m_q\/\\Lambda_{\\mathrm{QCD}}$. This stems from the fact that optical frequency measurements are all performed against the Cs hyperfine frequency, except $\\mathrm{Al}^{+}\/\\mathrm{Hg}^{+}$.\n\nSimilarly, we perform a global analysis for the variation with the gravitational potential exploiting all available comparisons as of October 2014 \\cite{peil2013, fortier2007, leefer2013} and the above Rb\/Cs and Sr\/Cs results.\nThe least-squares fit to these results yields independent constraints for the three couplings to gravity given in the second row of the Table in fig.~ \\ref{Graphdrift}.\n\nThe number of atomic systems contributing to improve these tests will continue to grow, e.g. with $^{88}$Sr$^+$ \\cite{Madej2012}\\cite{Barwood2014} and $^{171}$Yb \\cite{Lemke2009a}, thanks to the steady efforts of many laboratories worldwide in the field of optical frequency metrology.\n\n\n\n\\section{Advanced timekeeping}\\label{sec:timekeeping}\n\n\\textit{TAI calibration with atomic fountains}-- The International Atomic Time (TAI) which is based on approximately 400 atomic clocks, now gets its accuracy from some ten atomic fountain clocks worldwide (see e.g. \\cite{cras2014Petit}). In the last decade, the number of calibrations of TAI with atomic fountains has grown from approximately 10 per year to 4 to 6 per month in 2014, while simultaneously the accuracy improved from several $10^{-15}$ to a few $10^{-16}$, improving TAI a lot. Combining a tremendous number of monthly calibrations and a high accuracy, LNE-SYRTE atomic fountains are providing the largest contribution to the accuracy of TAI. Between 2007 and August 2014, they provided 197 calibrations out of a total of 407 calibrations worldwide, a weight of nearly 50\\%. fig.~\\ref{TAI_UTC}, Top shows these calibrations as published in \\emph{Circular T}, and the SI second (red line) which is the average over all primary calibrations computed by the BIPM on a monthly basis. This illustrates how research on laser-cooled atomic fountain started 25 years ago led to improving an important service and infrastructure for science and society.\n\n\n\n\n\\textit{UTC(OP): timescales using atomic fountain clocks}-- The UTC(OP) timescale, elaborated at SYRTE, in Observatoire de Paris, is the real time realization of UTC for France. It is a continuously operated time reference used for multiple purposes:\ndefinition of legal time disseminated in France, reference provided to French laboratories for\nsynchronization applications, pivot for French contributions to TAI, test of advanced time transfer\nmethods, link to UTC of the EGNOS system, contribution to the development of GALILEO, time reference for the ground-segment of the PHARAO\/ACES space mission \\cite{laurent2006, cacciapuoti2007,cacciapuoti2009}.\n\nProgress in the accuracy and most importantly in the reliability of SYRTE atomic fountains (see sect.~\\ref{fountains}) enabled a new implementation of UTC(OP). A new UTC(OP) algorithm based on a hydrogen maser steered by the atomic fountains was developed and\nimplemented in October 2012. The maser is predictable enough to reach a stability of $\\sim 10^{-15}$. The\natomic fountains allow the maser frequency to be calibrated with an uncertainty in the $10^{-16}$ range \\cite{guena2012, guena2014}. These features are sufficient to maintain a phase deviation of a few ns over $1-\n2$ months, which corresponds to the delay of publication of the BIPM \\emph{Circular T}.\nThis timescale is as autonomous and independent as possible, except a small long term steering to remain\nclose to UTC, and does not rely on any other timescale available in real time such as GPS time or\nother UTC(k). A timescale with these characteristics provides a powerful tool to understand current limits and eventually improve international timekeeping.\n\nPractically, UTC(OP) is realized using a microphase stepper fed by the reference maser. A frequency\ncorrection is updated every day to compensate the maser frequency and\nmaintain UTC(OP) close to UTC. This correction is the sum of two terms. The main term corresponds to the current frequency of the maser as measured\nby the fountains. The value is estimated with a linear extrapolation of the data covering the past 20~days\nto remain robust against possible interruptions of data provision or of the automatic data processing. The\nsecond term is a fine steering to maintain UTC(OP) close to UTC, compensating the\nfrequency and phase offset between UTC(OP) and UTC. It is updated monthly at the BIPM \\emph{Circular~T}\npublication. The steering correction is usually of the order of $10^{-15}$ or below.\n\nFigure \\ref{TAI_UTC}, bottom presents the comparison of three UTC(k) to UTC as published in \\emph{Circular~T}\nsince the implementation of the new UTC(OP). Over this period, UTC(OP) is one of the 3 best real time\nrealizations of UTC \\cite{rovera2013, abgrall2014}, with UTC(PTB), the pivot of time\ntransfers for international contributions to TAI, and UTC(USNO), the laboratory providing the largest number of clock data included in EAL computation. Departure between UTC(OP) and UTC remains well below 10~ns, with a rms value less than 3~ns. This is an improvement of about a factor of 5 compared to the previous realization method of the timescale. On-going instrumental upgrades shall further improve the short term stability of the timescale and the robustness of the system.\n\n\n\\begin{figure}[htb]\n\n\\includegraphics[width=0.45\\textwidth]{Fig4_GTAI4.pdf}\n\\includegraphics[width=0.45\\textwidth]{Fig4_GUTCk5.pdf}\n\\caption{Top: Calibrations of TAI by the atomic fountain PFSs. Filled symbols: contributions of SYRTE fountains. Solid red line: the SI.  Bottom: Comparisons of 3 UTC(k) to UTC: UTC(OP), UTC(PTB) and UTC(USNO). The inset shows the significant improvement achieved with the new method for generating UTC(OP) implemented at MJD 56218, compared to the previous one using a commercial Cs clock manually steered towards UTC.}\n\\label{TAI_UTC}\n\\end{figure}\n\n\\section{Toward a redefinition of the SI second}\n{}\n\nSeveral optical frequency standards are now largely surpassing Cs atomic fountains which realize the second of the international system of units (SI) and define the accuracy of TAI. This opens the inviting prospect of a redefinition the SI second. In 2001, anticipating this situation, the Consultative Committee for Time and Frequency (CCTF) of the Comit\\'e International des Poids et Mesures (CIPM) recommended that a list of Secondary Representations of the Second be established. Secondary Representations of the SI Second (SRS) are transitions which are used to realize frequency standards with excellent uncertainties, and which are measured in the SI system with accuracies close to the limit of Cs fountains. They are part of the broader list of recommended values of standard frequencies produced and maintained by the CCL-CCTF Working Group and adopted by the CIPM. Producing and maintaining these lists of recommended values is a vehicle to keep track of measurements providing the most stringent connections between the optical domain and the SI second, and to verify the level of consistency between these measurements. This is an important task to prepare for a possible redefinition of the SI second.\n\n\n\\textit{Contributions to the list of recommended values}--\nLNE-SYRTE provided several high accuracy absolute frequency measurements which contributed to the list of recommended values. The $^{87}$Rb hyperfine transition was measured repeatedly against Cs fountains, as already shown in fig.~\\ref{Graphdrift}. This transition became the first Secondary Representation of the Second proposed by the CCTF in 2004, based our early measurements, and adopted by the CIPM in 2006. After further significant progress visible in fig.~\\ref{Graphdrift}, the recommended value was revised by the 2012 CCTF and adopted in 2013. The origin of the vertical scale of the graph is the recommended value of 2012 and the gray area represents the recommended uncertainty. The weighted average of all data points (green line) gives $(2.15\\pm 1.48)\\times 10^{-16}$ is consistent with zero within the smallest overall uncertainty of the measurements ($4.4\\times 10^{-16}$). LNE-SYRTE also provided absolute frequency measurements that contributed to the establishment of the recommended values for the $^1S_0\\rightarrow{}^3P_0$ of $^{87}$Sr \\cite{Baillard2007b}, $^{88}$Sr \\cite{Baillard2007} and $^{199}$Hg \\cite{McFerran2012}. Using the transportable Cs fountain primary standard FOM, LNE-SYRTE also contributed to the establishment of the recommended value for H(1S-2S) (measured at the  Max Planck Institut f\\\"ur Quantenoptik, Garching, Germany) \\cite{parthey2011} and $^{40}$Ca$^{+}$ (at the University of Innsbruck, Austria) \\cite{chwalla2009}.\n\n$^{87}$Sr is currently the most widespread optical frequency standard. For this reason a large number of groups measured the absolute frequency of $^{87}$Sr against Cs with a remarkable degree of consistency, as can be seen in fig.~\\ref{Graphdrift}, middle. These measurements led the 2006 CCTF to recommend the $^{87}$Sr as a Secondary Representation of the SI second. The recommended value was updated by the 2012 CCTF based on measurements from 5 institutes. It has a recommended uncertainty of $1\\times 10^{-15}$. More recently, LNE-SYRTE reported a measurement of the $^{87}$Sr clock transition with an uncertainty of $3.1\\times 10^{-16}$ limited by the accuracy of atomic fountains \\cite{letargat2013}. This is the most accurate absolute measurement to date of any atomic frequency. One of the key factors for the measurement is the record stability between an optical and a microwave clock: $4.1\\times 10^{-14}\/\\sqrt{\\tau}$ against Cs (and $2.8\\times 10^{-14}\/\\sqrt{\\tau}$ against Rb). In 2014, PTB reported another absolute frequency measurement with an uncertainty of $3.9\\times 10^{-16}$ \\cite{falke2014}. These two last measurements are in excellent agreement.\n\n\n\n\n\\textit{Using a Secondary Representation of Second to calibrate TAI}-- One major application of primary frequency standards is to calibrate and steer the scale interval of the widely used International Atomic Time TAI. It is important to anticipate how a possible redefinition of the second would impact the elaboration of TAI. We used the FO2-Rb fountain to investigate how a Secondary Representation of Second could participate to TAI. Calibrations of the frequency of our reference hydrogen maser were produced with FO2-Rb, in a similar way that absolute calibrations are done with primary frequency standards. These data were then submitted to the BIPM and to the Working Group on Primary Frequency Standards. Following this submission, the BIPM and the Working Group defined how frequency standards based on Secondary Representations will be handled by the BIPM and how they will be included into the \\textit{Circular~T}. The Working Group was renamed Working Group on Primary and Secondary Frequency Standards and it was decided that calibrations produced by LNE-SYRTE with FO2-Rb could be included into \\textit{Circular~T} and, since July 2013, contribute to steering TAI. This was the first time that a transition other than the Cs hyperfine transition was used to steer TAI \\cite{guena2014}.\n\n\\textit{Absolute frequency measurement against the TAI ensemble}-- More than 40 formal calibrations of TAI with FO2-Rb have been sent, processed by the BIPM and published into \\textit{Circular~T}. These data can be used to measure the frequency Rb hyperfine transition directly against the second as realized by the TAI ensemble. This can be done with a statistical uncertainty of 1 part in $10^{-16}$, and therefore at the accuracy limit of primary frequency standards defining the scale interval of TAI \\cite{guena2014}. This illustrates how TAI provides worldwide access to the accuracy of Cs fountains. This also shows how recommended values of Secondary Representation of the Second based on optical transitions could be checked against the SI second as realized by the TAI ensemble.\n\n\n\n\\section{Prospects}\n\nIn the future, highly accurate atomic clocks and their applications will keep improving at a high pace. An important milestone in the field will be the simultaneous availability of advanced timescales, of the new generation of optical clocks and of the means to compared them remotely at unprecedented levels of uncertainty. In the coming decade, the ACES mission will allow ground-to-space comparisons to the $10^{-16}$ level and ground-to-ground comparisons to the mid $10^{-17}$ level \\cite{cras2014onACES}. Optical fiber links will allow comparisons of the most accurate optical clocks at their limit: $10^{-18}$ or better \\cite{cras2014onFiberLinks1,cras2014onFiberLinks2}. We can confidently predict major improvements in all applications of highly accurate atomic clocks. Availability of clocks and clock comparisons at the $10^{-18}$ level can further enable new applications. Clock comparisons determine Einstein's gravitational redshift between the 2 remote clock locations. For a clock at the surface of the Earth, $10^{-18}$ corresponds to an uncertainty of 1~cm in height-above-geoid, making the idea of clock-based geodesy realistic and potentially useful. Highly accurate clocks could become a new type of sensors for applications in Earth science, illustrating once again the fertilizing power of the quest for ever increased accuracy.\n\n\n\n\\section*{Acknowledgements}\n\nSYRTE is UMR CNRS 8630 between Centre National de la Recherche Scientifique (CNRS), Universit\\'e Pierre et Marie Curie (UPMC) and Observatoire de Paris. LNE, Laboratoire National de M\\'etrologie et d'Essais, is the French National Metrology Institute.\nSYRTE is a member of IFRAF, of the nanoK network of the R\\'egion \\^Ile de France and of the FIRST-TF LabeX. We acknowledge the large number of contributions of SYRTE technical services. This work is supported by LNE, CNRS, UPMC, Observatoire de Paris, IFRAF, nanoK, Ville de Paris, CNES, DGA, ERC AdOC, EMRP JRP SIB55 ITOC, EMRP JRP EXL01 QESOCAS. We are grateful to the University of Western Australia and to M.E.~Tobar for the long-lasting collaboration which gives us access to the cryogenic sapphire oscillator used in the LNE-SYRTE ultra-stable reference.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}