diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzpahc" "b/data_all_eng_slimpj/shuffled/split2/finalzzpahc" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzpahc" @@ -0,0 +1,5 @@ +{"text":"\n\n\n\\section{Introduction}\n\\color{black}\n On a contact manifold there exists a natural class of flows, the so-called Reeb flows. Although the dynamics of distinct Reeb flows on the same contact manifold can be very different, there are dynamical properties which are common to all Reeb flows on a given contact manifold. For instance, the combined works of Hofer \\cite{Hofer} and Taubes \\cite{Taubes} imply that on a closed contact 3-manifold all Reeb flows have at least one periodic orbit. \n In this paper we construct a large class of contact manifolds on which all Reeb flows have chaotic dynamics. Surprisingly, some of the contact manifolds we construct have a very simple topology, which contrasts with the complicated dynamics of their Reeb flows.\n \n \n\nA contact structure is said to have positive entropy if all Reeb flows associated to this contact structure have positive topological entropy.\nWe show that there exist contact structures with positive entropy on spheres of dimension $\\geq7$ and on $S^3\\times S^2$. As a consequence we prove that every manifold of dimension $\\geq 7$ that admits an exactly fillable contact structure also admits a (possibly different) contact structure with positive entropy. \\color{black} \nOur approach to prove these results is based on wrapped Floer homology and uses in an essential way its product structure. This product structure enables us to define the notion of algebraic growth of wrapped Floer homology, and we relate this growth to the volume growth of Reeb flows. Even though the richer algebraic structures in Floer homology were studied extensively, so far they lead to only very few applications in dynamics: the ones we are aware of are Viterbo's result \\cite{Viterbo1999} on the existence of one closed Reeb orbit on hypersurfaces of restricted contact type in Liouville domains with vanishing symplectic homology, and Ritter's result \\cite{Ritter2013} on the existence of Reeb chords for exactly fillable Legendrian submanifolds on Liouville domains with vanishing symplectic homology.\n\n\n\\subsection{Basic notions}\n\nAn important measure of the complexity of a dynamical system on a manifold $M$ is the topological entropy $h_{top}$ which quantifies in a single number the exponential complexity of the system. We refer the reader to \\cite{Hasselblat-Katok} for the definition and basic properties of $h_{\\mathrm{top}}$. \nBy deep results of Yomdin and Newhouse, $h_{\\mathrm{top}}(\\phi)$ for a $C^{\\infty}$-flow $\\phi = (\\phi^t)_{t\\in\\R}$ equals the exponential growth rate of volume\n\\begin{equation*}\nv(\\phi) = \\sup_{N\\subset M} v(\\phi,N), \\text{ where }\n\\end{equation*}\n\\begin{equation} \\label{Yomdin}\nv(\\phi, N) = \\limsup_{t\\to \\infty} \\frac{\\log \\Vol_g^n(\\phi^t(N))}{t}.\n\\end{equation}\nHere, $n = \\dim N$, the supremum is taken over all submanifolds $N \\subset M$, and $\\Vol^n_g$ is the $n$-dimensional volume with respect to some Riemannian metric $g$ on $M$. \n\nIn this paper we study the topological entropy for Reeb flows of contact manifolds.\nRecall that a \\textit{(co-oriented) contact manifold} $(\\Sigma, \\xi)$ is \na compact odd-dimensional manifold $\\Sigma^{2n-1}$ equipped with a contact structure $\\xi$, that is, a hyperplane distribution on $\\Sigma$ which is given by $\\xi = \\ker \\alpha$ for a $1$-form $\\alpha$ with $\\alpha \\wedge (d\\alpha)^{n-1} \\neq 0$. Such an $\\alpha$ is called a \\textit{contact form} on $(\\Sigma,\\xi)$, and we can associate to it the \\textit{Reeb vector field} $X_{\\alpha}$ defined by $\\iota_{X_{\\alpha}}d\\alpha = 0$, $\\alpha(X_{\\alpha})= 1$. Denote the flow of $X_\\alpha$, the \\textit{Reeb flow} of $\\alpha$, by $\\phi_{\\alpha}= (\\phi_{\\alpha}^t)_{t\\in \\R}$. \nAn \\textit{isotropic submanifold} of $\\Sigma^{2n-1}$ is one whose tangent space is contained in $\\xi$; isotropic submanifolds of dimension $n-1$ are called \\textit{Legendrian submanifolds}. \n\n\\subsection{Main results}\nThe main result of this paper is the existence of contact structures with positive entropy on high dimensional manifolds.\n\n\n \\begin{thm}\\label{spheres}\n \\\n \n \\begin{itemize} \n \\item [A)]\\label{itm:spheres}\n Let $S^{2n-1}$ be the $(2n-1)$ - dimensional sphere with its standard smooth structure.\nFor $n\\geq 4$ there exists a contact structure on $S^{2n-1}$ with positive entropy.\n\\item [B)] \\label{itm:S3S2}\nThere exists a contact structure on $S^3\\times S^2$ with positive entropy.\n\\end{itemize}\n\\end{thm}\n\nRecall that a contact manifold is said to be exactly fillable if it is the boundary of a Liouville domain. From Theorem \\ref{spheres} and the methods developed in this paper we obtain the following more general result.\n\\begin{thm}\\label{maincorollary}\n\\\n\\begin{itemize}\n\\item [$\\clubsuit$] \n\\ If $V$ is a manifold of dimension $2n-1\\geq 7$ that admits an exactly fillable contact structure, then $V$ admits a contact structure with positive entropy.\n\\item [$\\diamondsuit$]\nIf $V$ is a $5$-manifold that admits an exactly fillable contact structure, then the connected sum $V \\# (S^3\\times S^2)$ admits a contact structure with positive entropy.\n\\end{itemize}\n\\end{thm}\n\nNote that the standard contact structure on spheres as well as the canonical contact structure on $S^{*}S^3 \\cong S^3 \\times S^2$ have a contact form with periodic Reeb flow. In particular these are not diffeomorphic to the contact structures in Theorem \\ref{spheres}. Other exotic contact spheres have been constructed by several authors, see \\cite{Eliashberg1991, Ustilovsky1999, GeigesDing2004, McLean2011}. The contact spheres constructed in this paper are, from our perspective, the ``most exotic'' ones. From the dynamical point of view they are the most remote from the standard contact spheres since they admit Legendrian submanifolds that have exponential volume growth under every Reeb flow. It would be interesting to relate our examples of exotic contact spheres to others that were constructed so far. \n\nIn order to explain further the relevance of these results we recall what is known about the topological entropy of Reeb flows.\nMotivated by results on topological entropy for geodesic flows (see \\cite{Paternain}), combined with the geometric ideas of \\cite{FrauenfelderSchlenk2006}, Macarini and Schlenk proved in \\cite{MacariniSchlenk2011} that for various manifolds $Q$ the unit cotangent bundle $(S^{*}Q, \\xi)$ equipped with the canonical contact structure $\\xi$ has positive entropy\\footnote{In a recent work \\cite{Dahinden} Dahinden extended the results in \\cite{MacariniSchlenk2011} proving that on the unit cotangent bundles $(S^{*}Q, \\xi)$ studied in \\cite{MacariniSchlenk2011} every positive contactomorphism has positive topological entropy. It would be interested to investigate if Dahinden's result is true for any contact manifold with positive entropy.}.\n\nIn previous works of the first author, different examples of contact 3-manifolds with positive entropy were discovered. In \\cite{Alves-Cylindrical,Alves-Anosov,Alves-Legendrian} it was shown that contact 3-manifolds with positive entropy exist in abundance: there exist hyperbolic contact 3-manifolds with positive entropy (see also \\cite{ACH}), non-fillable contact 3-manifolds with positive entropy, and even 3-manifolds which admit infinitely many non-diffeomorphic contact structures with positive entropy. This shows that the class of contact manifolds with positive entropy is much larger than the class of unit cotangent bundles over surfaces with positive entropy, which were studied in \\cite{MacariniSchlenk2011}.\nOne common feature of all known examples of contact 3-manifolds with positive entropy is that the fundamental group of the underlying smooth 3-manifold has exponential growth. We expect this to be always the case:\n\\begin{conj} \\label{conjecture3dim}\nIf a contact 3-manifold $(\\Sigma,\\xi)$ has positive entropy, then $\\pi_1(\\Sigma)$ grows exponentially.\n\\end{conj}\n\nAlready from the unit cotangent bundles of simply connected rationally hyperbolic manifolds, which were considered in \\cite{MacariniSchlenk2011}, we know that Conjecture \\ref{conjecture3dim} is false in higher dimensions.\nHowever it is natural to ask if there are restrictions on the smooth topology of contact manifolds with positive entropy in higher dimensions.\nTheorem \\ref{spheres} shows that in contrast to what happens in dimension three, the phenomenon in higher dimensions is quite flexible from the topological point of view.\n\n\\color{black}\n\\begin{rem} {Examples of contact manifolds of dimension $\\geq 9$ which have positive entropy and are not unit cotangent bundles are also constructed using connected sums in an ongoing work of the first author and Macarini \\cite{connected}, following an idea of Schlenk. However, these contact manifolds have very complicated smooth topology, in the sense that the underlying smooth manifolds are rationally hyperbolic. For this reason they are much less surprising than the ones obtained in the present paper.} \\end{rem}\n\\color{black}\n\nLet us now explain our approach to establishing these results.\n\n\n\\subsection{Symplectic and algebraic growth}\n\nTo establish our results we introduce the notion of algebraic growth of wrapped Floer homology. This notion is useful because, on one hand, it gives a lower bound for the growth rate of wrapped Floer homology defined using its action filtration and, on the other hand, it is stable under several geometric modifications of Liouville domains.\n\n The contact manifolds we consider in this paper arise as boundaries of Liouville domains. Recall that a Liouville domain $M=(Y,\\omega, \\lambda)$ is an exact symplectic manifold $(Y,\\omega)$ with boundary $\\Sigma = \\partial Y$ and a primitive $\\lambda$ of $\\omega$ such that $\\alpha_M := \\lambda|_{\\Sigma}$ is a contact form on $\\Sigma$: we let $ \\xi_{M}= \\ker \\alpha_M$ be the contact structure induced by $M$ on $\\Sigma$. For two exact Lagrangians $L_0$ and $L_1$ in $M$ that are asymptotically conical, i.e. conical near $\\partial Y$ with Legendrian boundaries $\\Lambda_0$ and $\\Lambda_1$ in $(\\Sigma,\\xi_M)$, we consider the wrapped Floer homology of $(M,L_0,L_1)$ with $\\Z_2$-coefficients denoted by $\\mathrm{HW}(M,L_0 \\to L_1)$, whose underlying chain complex is, informally speaking, generated by Reeb chords from $\\Lambda_0$ to $\\Lambda_1$ and intersections of $L_0$ and $L_1$. We write $\\mathrm{HW}(M,L)$ for $\\mathrm{HW}(M,L \\to L)$, see Section \\ref{subsec:def}.\n\nResults on positive entropy can be obtained from the exponential \\textbf{ symplectic growth} of wrapped Floer homology, which is defined as follows. \\color{black}\nBy considering only critical points below an action value $a$, one obtains the filtered Floer homology $\\mathrm{HW}^a(M,L_0 \\to L_1)$. \n The homologies $\\mathrm{HW}^a(M,L_0 \\to L_1)$ form a natural filtration of $\\mathrm{HW}(M,L_0 \\to L_1)$, and they come with natural maps $\\iota_a: \\mathrm{HW}^a(M,L_0 \\to L_1) \\rightarrow \\mathrm{HW}(M,L_0 \\to L_1)$ into the (unfiltered) Floer homology. The \\textit{exponential symplectic growth rate} ${\\Gamma}^{\\mathrm{symp}}(M,L_0 \\to L_1)$ of $\\mathrm{HW}(M,L_0 \\to L_1)$ is given by\n\\begin{align}\\label{Gamma_symp}\n\\Gamma^{\\mathrm{symp}}(M,L_0 \\to L_1) = \\limsup_{a \\to \\infty} \\frac{\\log (\\dim \\mathrm{Im} \\ \\iota_a)}{a};\n\\end{align}\nsee Section \\ref{subsec:def} and Definition \\ref{defi:Gamma_symp}.\nSince the generators of $\\mathrm{HW}(M,L_0 \\to L_1)$ correspond essentially to Reeb chords from $\\Lambda_0$ to $\\Lambda_1$, the symplectic growth gives a lower bound on the growth of Reeb chords with respect to their action. \\color{black}\nAssuming that $\\Lambda_1$ is a sphere, we adapt the ideas of the first author in \\cite{Alves-Legendrian} to get lower bounds for the volume growth $v(\\phi_{\\alpha},\\Lambda_0)$ in terms of the exponential symplectic growth rate of $\\mathrm{HW}(M,L_0,L_1)$ for every contact form $\\alpha$ on $\\xi_{M}$. \n\n\nA \\textit{topological operation} on a Liouville domain $M$ is a recipe for producing a new Liouville domain $N$ from $M$. \nTo obtain examples of contact manifolds with positive entropy we perform certain topological operations on Liouville domains. The operations we consider are: attaching symplectic handles on $M$ and, in the case $M$ is the unit disk bundle of a manifold, plumbing $M$ with the unit disk bundle of another manifold. \nAlthough one can understand the change or invariance of the (unfiltered) wrapped Floer homology under these operations, it is often much harder or not even possible to understand the effect of these operations on the symplectic growth. \nFor instance, by an adaptation of a theorem of Cieliebak \\cite{Cieliebak2001} we show that $\\mathrm{HW}({M}',L)$ is isomorphic to $\\mathrm{HW}(M,L\\cap M)$ if ${M}'$ is obtained by subcritical handle attachment on $M$ (Theorem \\ref{Viterbo_iso}). By contrast it is much harder to control the filtered Floer homology under this operation, see \\cite{McLean2011} for an approach in the case of symplectic homology. In the case of plumbings of two cotangent bundles the computational results of a relevant part of the unfiltered wrapped Floer homology obtained by \\cite{AbouzaidSmith2012} do not carry over to the symplectic growth rates of the plumbing.\n\nTo overcome this difficulty we look at a notion of growth that is defined purely in terms the algebraic structure on wrapped Floer homology, the \\textbf{algebraic growth}. \nLet us explain this briefly. \nLet $A$ be a (not necessarily unital) $K$-algebra with multiplication $\\star$ and $S \\subset A$ a finite set of elements of $A$. \\color{black} \nGiven $j\\geq 0$, let $N_S(j) = \\{ a \\in A \\, \\mid \\, a = s_1\\star s_2 \\star \\cdots \\star s_j; \\, s_1, \\dots, s_j \\in S \\}$; i.e.\\ $N_S(j)$ is the set of elements of $A$ that can be written as a product of $j$ not necessarily distinct elements of $S$.\n We define $W_S(n) \\subset A$ to be the smallest $K$-vector space that contains the union $\\bigcup_{j=1}^{n} N_S(j)$. \\color{black} \n The \\textit{exponential algebraic growth rate} of the pair $(A,S)$ is defined as \\color{black}\n\\[\n\\Gamma^{\\mathrm{alg}}_S(A) = \\limsup_{n\\to \\infty}\\frac{1}{n}\\log\\dim_K W_S(n) \\in [0,\\infty).\n\\]\n\\color{black}\n In case $A = K \\langle G \\rangle$ is the group algebra over a finitely generated group $(G = \\langle S \\rangle ,\\star) $,\nit is elementary to see that $\\Gamma^{\\mathrm{alg}}_S(A)$ coincides with the exponential algebraic growth of $G$ in the usual geometric group theoretical sense.\nNow, induced by the triangle product in Floer homology, $\\mathrm{HW}(M,L)$ is equipped with a ring structure $\\star$ turning it into a $\\Z_2$-algebra. \\color{black} Given a finite set $S$ of $\\mathrm{HW}(M,L)$ we define (cf.\\ Definition \\ref{defi:growthHW}) $\\Gamma_{S}^{\\mathrm{alg}}(M,L):= \\Gamma_{S}^{\\mathrm{alg}}(\\mathrm{HW}(M,L)) $.\n We say that $\\mathrm{HW}(M,L)$ has \\textit{exponential algebraic growth} if there exists a finite subset $S$ of $\\mathrm{HW}(M,L)$ such that $\\Gamma_{S}^{\\mathrm{alg}}(M,L) > 0$. \\color{black}\n\n\nOur main motivation for studying the exponential algebraic growth of $\\mathrm{HW}$ is the following\n\\begin{prop}\\label{prop:operations}\nLet $M$ be a Liouville domain and $L$ be an asymptotically conical exact Lagrangian in it, and assume that $\\mathrm{HW}(M, L)$ has exponential algebraic growth. Then we have:\n\\begin{itemize}\n\\item [A)] The Liouville domain ${M}'$ obtained by attaching subcritical handles to $M$ has exponential algebraic growth of $\\mathrm{HW}$. More precisely, if the attachments are made away from $L$ (so that $L$ survives as an asymptotically conical exact Lagrangian submanifold of ${M}'$) then $\\mathrm{HW}({M}',L)$ has exponential algebraic growth. \n\\item [B)] If $M$ is the unit disk bundle of a closed orientable manifold $Q^n$ whose fundamental group grows exponentially, and ${M}'$ is obtained by a plumbing whose graph is a tree and one of the vertices is $M$, then ${M}'$ has exponential algebraic growth of $\\mathrm{HW}$. More precisely, if $L_q$ is a unit disk fibre in $M$ and the plumbing is done away from $L_q$ then $\\mathrm{HW}({M}',L_q)$ has exponential algebraic growth.\n\\end{itemize}\n\\end{prop}\nThis result essentially says that plumbing and subcritical surgeries are topological operations that preserve exponential algebraic growth of $\\mathrm{HW}$, and will allow us to construct many examples of Liouville domains which admit asymptotically conical exact Lagrangian disks with exponential algebraic growth of $\\mathrm{HW}$.\n\n\nThe exponential algebraic growth of our examples stems from the algebraic growth of the homology of the based loop space $H_{*}(\\Omega Q)$ equipped with the Pontrjagin product, where $Q$ is a compact manifold. In fact, we will only use the degree $0$ part whose algebraic growth is that of $\\pi_1(Q)$. \n\n\\begin{rem}\nThe exponential algebraic growth of symplectic homology always vanishes since its product is commutative. Thus our approach is specifically designed for the open string case. \n\\end{rem}\n\n\nIn order to obtain our main results we will bound the topological entropy of Reeb flows from below in terms of the algebraic growth of $\\mathrm{HW}(M,L)$. For that we will use the crucial fact that the spectral number $c: \\mathrm{HW}(M,L) \\rightarrow \\R_+$ defined by $c(x) = \\inf\\{a \\in \\R \\, |\\, x \\in \\mathrm{Im} \\, i_a \\}$ is subadditive, i.e. $c(x \\star y) \\leq c(x) + c(y)$ for all $x,y \\in \\mathrm{HW}(M,L)$. \\color{black} It follows (see Proposition~\\ref{prop:alg_symp}) that for any finite $ S \\subset \\mathrm{HW}(M,L)$ we have \n\\begin{equation*}\\label{algebraic_symplectic}\n\\Gamma^{\\mathrm{symp}}(M,L) \\geq \\frac{1}{\\rho(S)}\\Gamma^{\\mathrm{alg}}_S(M,L),\n\\end{equation*}\nwhere $\\rho(S) = \\max_{s\\in S} c(s)$. \nBy using that $\\mathrm{HW}(M,L \\to L_1)$ is a module over $\\left(\\mathrm{HW}(M,L), \\star \\right)$, this lower bound can be extended to $\\Gamma^{\\mathrm{symp}}(M,L \\to L_1)$ for all $L_1$ that are exact Lagrangian isotopic to $L$, see Lemma \\ref{lemmaprelim}. In other words, exponential algebraic growth of $\\mathrm{HW}(M,L)$ implies positive symplectic growth of $\\mathrm{HW}(M,L \\to L_1)$. This, combined with ideas from \\cite{Alves-Legendrian}, leads to \n\\begin{thm} \\label{theorementropy}\nLet $L$ be an asymptotically conical exact Lagrangian on a Liouville domain $M=(Y,\\omega,\\lambda)$, $\\Sigma:= \\partial Y$ and $ \\alpha_{M}:= \\lambda|_\\Sigma$. We denote by $\\xi_{M}:=\\ker \\alpha_{M})$ the contact structure induced by $M$ on $\\Sigma$.\nAssume that there is a finite set $ S \\subset \\mathrm{HW}(M, L)$ such that $\\Gamma^{\\mathrm{alg}}_S(M,L) >0$ and that $\\Lambda = \\partial L$ is a sphere. Then, for every contact form $\\alpha$ on $(\\Sigma ,\\xi_{M})$ the topological entropy of the Reeb flow $\\phi_{\\alpha}$ is positive. Moreover, if $\\mathsf{f}_{\\alpha}$ is the function such that $\\mathsf{f}_{\\alpha}\\alpha_{M} = \\alpha$ then\n\\begin{equation*}\nh_{\\mathrm{top}}(\\phi_{\\alpha}) \\geq \\frac{\\Gamma^{\\mathrm{alg}}_S(M,L)}{\\rho(S) \\max(\\mathsf{f}_{\\alpha})}.\n\\end{equation*}\n\\end{thm} \\color{black}\n\n\nOur paper is organised as follows.\nIn section \\ref{sec:Floer} we consider the algebraic growth and the growth of filtered directed systems in general, and then we recall the definition of wrapped Floer homology together with its product structure. In section \\ref{sec:Viterbo} we present the construction of the Viterbo map and derive some of its properties. Section \\ref{sec:entropy} establishes implications of the growth properties of $\\mathrm{HW}$ to topological entropy. In section \\ref{sec:ring_module} we recall the computation of the algebra structure of the Floer homology of unit disk bundles and in section \\ref{sec:top_operations} we give a proof of the invariance of $\\mathrm{HW}$ under subcritical handle attachment, recollect a result on $\\mathrm{HW}$ of plumbings and prove Proposition \\ref{prop:operations}. Finally, in section \\ref{sec:constructions}, we construct our examples and prove the main theorems. The Appendix contains a construction of exact Lagrangian cobordisms used in the paper. \n\n\\acknowledgementname: Most of this work was done when the second author visited the Universit\\'e of Neuch\\^atel supported by the Erasmus mobility program, and the first author visited the Universit\\\"at M\\\"unster supported by the SFB\/TR 191. This work greatly benefited from discussions with Felix Schlenk and Peter Albers: we thank them for their interest in this work and their many suggestions. We also thank Lucas Dahinden for carefully reading the manuscript. \n\n\n\\section{Wrapped Floer homology and its growth}\\label{sec:Floer}\nAs explained in the introduction, two features of wrapped Floer homology are crucial in this paper. \n\n First, its natural filtration by action gives the wrapped Floer homology $\\mathrm{HW}$ the structure of a filtered directed system and allows one to define the spectral value of elements of $\\mathrm{HW}$. These give rise to the notion of symplectic growth\\footnote{This was explicitly observed in \\cite{Mclean2015} although it is implicit in \\cite{FrauenfelderSchlenk2006,MacariniSchlenk2011}.} of $\\mathrm{HW}$; this is explained in Section \\ref{sec:wrapped}. \n \n Second, the product structure of $\\mathrm{HW}$ gives it the structure of an algebra and gives rise to the notion of algebraic growth of $\\mathrm{HW}$. This is explained in Section \\ref{sec:productonHW}. \nThe link between these notions is given by the crucial fact that the spectral number is subadditive with respect to the product structure on $\\mathrm{HW}$, see also Section \\ref{sec:productonHW}.\n\n We first recall the relevant algebraic notions and deduce some direct consequences.\n \n\n\\subsection{Algebraic growth and growth of filtered directed systems} \nFix a field $K$. We use the convention that $\\log(0) := 0$. \n\n\\subsubsection{Filtered directed systems and growth}\\label{subsubsec:fds}\n\n\\begin{defn}\nA \\textit{filtered directed system} over $\\R_{+} = [0,\\infty)$ or for short \\textit{f.d.s.} is a pair $(V,\\pi)$ where\n\\begin{itemize}\n\\item $V_t$, $t \\in [0,\\infty)$, are finite dimensional $K$-vector spaces.\n\\item $\\pi_{s \\rightarrow t}: V_s \\rightarrow V_t$, for $s\\leq t$ are homomorphisms (\\textit{persistence homomorphisms}), such that \n$\\pi_{s \\rightarrow t}\\circ\\pi_{r \\rightarrow s}=\\pi_{r \\rightarrow t}$ for $r\\leq s\\leq t$, and $\\pi_{t\\rightarrow t} = \\id_{V_t}$ for all $t \\in \\R_{+}$.\n\\end{itemize}\n\\end{defn}\n\n\\color{black}\n\n\n\\color{black}\nLet $\\mathfrak{J}$ be the smallest vector space of $\\bigoplus_{t\\in \\R_+} V_t$ containing $\\bigcup_{s \\leq t} \\{\\pi_{s\\rightarrow t}(x_s) - x_s\\}$. \nThe \\textit{direct limit} $\\dlim V$ of $V$ is defined by $\\dlim V := \\bigoplus_{t\\in \\R_+} V_t\/\\mathfrak{J}$. The inclusions $V_t \\hookrightarrow \\bigoplus_{t\\in \\R_+} V_t$ induce maps to $\\dlim V$ which we denote by $i_t$. \nThe \\textit{spectral number} $c_V$, or just $c$ if the context is clear, of an element $x \\in \\dlim V$ is\n\\[ \nc_V(x) := \\inf\\{t \\in [0,\\infty) \\, | \\, \\exists x_t \\in V_t \\text{ such that } i_t(x_t) = x \\}.\n\\]\nIt is clear from the definition of $c_V$ that if $x_1,..,x_n \\in V$ and $k_1,...,k_n\\in K$ we have\n\\begin{equation} \\label{spectralsuminequality}\nc_V\\bigg( \\sum_{i=1}^n k_i x_i \\bigg) \\leq \\max_{1 \\leq i \\leq n}{c_V(x_i)}.\n\\end{equation}\n\n\\begin{defn}\\label{fds_growth}\nLet $d_t^V := \\dim \\{x \\, | \\, c_V(x) \\leq t\\}$.\nThe \\textit{exponential growth rate} of the f.d.s. $V$ is\n\\[\n\\widetilde{\\Gamma}(V) := \\limsup_{t \\rightarrow \\infty}\\frac{1}{t} \\log d^V_t.\n\\]\nWe say that $V$ has exponential growth if $0 < \\widetilde{\\Gamma}(V) < \\infty$.\n\\end{defn}\n\n\n\\begin{defn}\nA \\textit{morphism} between f.d.s. $(V,\\pi)$ and $(V^{'},\\pi^{'})$ is a collection of homomorphisms \n$f = (f_t)_{t\\in[0, \\infty)}$, $f_t : V_t \\rightarrow V^{'}_t$, that are compatible with respect to the persistence homomorphisms: \n\\begin{equation}\\label{morph}\nf_t \\circ \\pi_{s \\rightarrow t} = \\pi^{'}_{s\\rightarrow t}\\circ f_s.\n\\end{equation}\nAn \\textit{asymptotic morphism} is a collection of homomorphisms $f_t : V_t \\rightarrow V^{'}_t$, $t\\in (K, \\infty)$, for some $K > 0$ such that \\eqref{morph} holds for $K0$. \nWe then get\n\\begin{equation*} \n\\begin{split}\n\\widetilde{\\Gamma}(V) &= \\limsup_{t \\rightarrow \\infty} \\frac{\\log \\, d_t^V}{t} \\leq \\limsup_{t \\rightarrow \\infty} \n\\frac{\\log \\, d^W_{t+c_W(m_0)}}{t} \\\\ &= \n\\limsup_{t \\rightarrow \\infty} \\frac{\\log \\, d^W_{t+c_W(m_0)}}{t +c_W(m_0)} \\frac{t+c_W(m_0)}{t} = \n\\widetilde{\\Gamma}(W).\n\\end{split}\n\\end{equation*}\nThis proves \\eqref{firsteq}. Inequality \\eqref{seceq} is obtained by combining \\eqref{firsteq} with Lemma \\ref{finite_growth}.\n\\qed\n\n\n\n\n\n\nIn order to get results on entropy, we will need the following notions.\n\n\\begin{defn}\nLet $\\mathcal{W} = W(i)_{i \\in I}$ be a family of f.d.s. with direct limits $M(i)$ that are modules over $A:= \\dlim V$.\n We say that the family $M(i)_{i \\in I}$ is \\textit{uniformly stretched} if there exists a constant $B\\geq 0$ such that for every $i \\in I$ there exists a stretching element $m_i \\in M(i)$ with $c_{M(i)}(m_i) \\leq B$. \n\\end{defn}\n \n\\begin{defn} \\label{defi:fam_growth}\nLet $\\mathcal{W} = W(i)_{i \\in I}$ be a family of filtered directed systems. The \\textit{uniform exponential growth rate } of $\\mathcal{W}$ is \n\\[\n\\widetilde{\\Gamma}_{i\\in I}(\\mathcal{W}) := \\limsup_{t \\rightarrow \\infty} \\frac{1}{t} \\log \\left(\\inf_{I} d_t^{W(i)}\\right).\n\\]\n\\end{defn}\n\n\\color{black}\n\\begin{lem}\\label{mod_fam_growth}\nLet $V$ be a f.d.s. such that $A= \\dlim V$ has a $K$-algebra structure with multiplication $\\star$. \nLet $\\mathcal{W} = W(i)_{i \\in I}$ be a family of f.d.s. such that for every $i\\in I$ the direct limit $M(i)=\\dlim W(i)$ is a module over $A$ with multiplication $\\ast(i)$. Assume that $c_V$ is subadditive with respect to $\\star$, that $c_{W(i)}$ is subadditive with respect to $\\ast(i)$ for every $i\\in I$, and that the family $M(i)_{i \\in I}$ is uniformly stretched over the algebra $A$.\nThen\n\\begin{equation}\n\\widetilde{\\Gamma}_{i\\in I}(\\mathcal{W}) \\geq \\widetilde{\\Gamma} (V).\n\\end{equation}\n\\end{lem} \\color{black}\n\\textit{Proof: }\nSince $M(i)_{i \\in I}$ is uniformly stretched there exists $B>0$ such that for every $i \\in I$, we can find a stretching element $m_i \\in M(i)$ with $c_{M(i)}(m_i) \\leq B$. Hence we have by \\eqref{subadd_module} that $d_t^V \\leq \\inf_{I}d^{W(i)}_{t+B}$ and the result is obtained as in the proof of Lemma \\ref{mod_growth}. \n\\qed\n\n\n\n \n\\subsection{Wrapped Floer homology}\\label{subsec:def}\n\nIn the following we give the definition and conventions for wrapped Floer homology used in this paper. This Floer type homology theory appeared in \\cite{AS-iso} for contangent bundles, and the case of general Liouville domains can be found in \\cite{AbouzaidSeidel2010}. We refer to these papers and \\cite[Section 4]{Ritter2013} for more details.\n\n\\subsubsection{Liouville domains and Lagrangians}\n\nA \\textit{Liouville domain} $M=(Y,\\omega, \\lambda)$ is an exact symplectic manifold $(Y, \\omega)$ with boundary $\\Sigma = \\partial Y$ and a primitive $\\lambda$ of $\\omega$ such that $\\alpha_M = \\lambda|_{\\Sigma}$ is a contact form on $\\Sigma$.\n The \\textit{Liouville vector field} X, is given by $i_X\\omega = \\lambda$ and points outwards along~$\\Sigma$. Using the flow of X, one can attach an infinite cone to $M$ along $\\Sigma$ that gives the \\textit{completion} $\\widehat{M}:=\\left(\\widehat{Y}, \\widehat{\\omega}, \\widehat{\\lambda}\\right)$ of $M$ with $\\widehat{Y} = Y \\cup_{\\Sigma} \\left([1, \\infty) \\times \\Sigma \\right)$, $\\widehat{\\lambda}|_{Y} = \\lambda$, $\\widehat{\\lambda}|_{[1, \\infty) \\times \\Sigma} = r\\alpha_M$, and $\\widehat{\\omega}=d \\widehat{\\lambda}$. \n\n\n\\begin{rem}In order to simplify notation, we will usually write $M$ and $\\widehat{M}$ instead of $Y$ and $\\widehat{Y}$, respectively, as the domain of Hamiltonian functions or the target space of Floer trajectories. This does not cause any confusion since the smooth manifolds $Y$ and $\\widehat{Y}$ are part of the data defining $M$ and $\\widehat{M}$, respectively. Similarly, when we write $\\Sigma=\\partial M$ it should be understood as $\\Sigma= \\partial Y$. \\end{rem}\n\nLet $\\mathsf{f}: \\partial Y \\rightarrow (0, \\infty)$ be a smooth function. Let $Y_{\\mathsf{f}} = \\widehat{Y} \\setminus \\{(r,x) \\, | \\, r > \\mathsf{f}(x), x \\in \\partial Y\\}$. It is easy to see that $M_{\\mathsf{f}} = (Y_{\\mathsf{f}},\\widehat{\\omega}|_{Y_{\\mathsf{f}}},\\widehat{\\lambda}|_{Y_{\\mathsf{f}}})$ is a Liouville domain. For example, given $\\delta >0$ we denote by $M_{1+\\delta}$ the Liouville domain $(Y_{1+\\delta},\\omega_{1+\\delta},\\lambda_{1+\\delta})$ embedded in $\\widehat{M}$ defined by ${Y}_{1+\\delta} = Y \\cup_{\\Sigma} \\left([1, 1+\\delta] \\times \\Sigma\\right)$, \n$\\omega_{1+\\delta} = \\widehat{\\omega}|_{Y_{1+\\delta}}$, $\\lambda_{1+\\delta} = \\widehat{\\lambda}|_{Y_{1+\\delta}}$. \n\n\nIn our paper we only consider Liouville domains that have vanishing first chern class $c_1(M) \\subset \\mathrm{H}^2(M;\\Z)$. \n\n\n\\ \n\nWe consider Lagrangians $(L, \\partial L)$ in $(M, \\Sigma)$ that are exact, i.e. $\\lambda|{L} = df$, and that satisfy \n\\begin{equation}\\label{asympt}\n\\begin{split}\n &\\Lambda = \\partial L \\text{ is a Legendrian submanifold in } (\\Sigma,\\xi_M), \\\\\n & L \\cap [1-\\epsilon, 1] \\times \\Sigma = [1-\\epsilon, 1] \\times \\Lambda \\text{ for a sufficiently small } \\epsilon >0 .\n\\end{split} \n\\end{equation}\nWe will call a Lagrangian that satisfies \\eqref{asympt} \\textit{asymptotically conical}. We can extend it naturally to an exact Lagrangian $\\widehat{L} = L \\cup_{\\Lambda} ([1,\\infty) \\times \\Lambda)$ in $\\widehat{M}$.\nWe will refer to a Lagrangian in $\\widehat{M}$ of this form also as \\textit{asymptotically conical (with respect to $M$)}. More generally, given a subset $U \\subset \\widehat{M}$ we say that $L$ is \\textit{conical in $U$} if the Liouville vector field is tangent to $L\\cap \\mathrm{int}(U)$ in the interior $ \\mathrm{int}(U)$ of $U$. \n\n\n\n\\subsubsection{Wrapped Floer homology} \\label{sec:wrapped}\n\n\nFor two asymptotically conical exact Lagrangians $L_0$ and $L_1$ in ${M}$ denote by $\\mathcal{P}_{L_0 \\to L_1} = \\{\\gamma : [0,1] \\rightarrow \\widehat{M} \\, |\\, \\gamma(0) \\in \\widehat{L}_0, \\, \\gamma(1) \\in \\widehat{L}_1\\}$\nthe space of (smooth) paths from $\\widehat{L}_0$ to $\\widehat{L}_1$. \n\nDenote by $X_{\\alpha_M}$ the Reeb vector field on the boundary $(\\Sigma, \\xi_{M} = \\ker{\\alpha_M})$. \nA \\textit{Reeb chord of length $T$} of $\\alpha_M$ from $\\Lambda_0 = \\partial L_0$ to $\\Lambda_1 = \\partial L_1$ is a map $\\gamma: [0,T] \\to \\Sigma$ with $\\dot{\\gamma}(t) = X_{\\alpha_M}(\\gamma(t))$ with $\\gamma(0) \\in \\Lambda_0$ and $\\gamma(T) \\in \\Lambda_1$. Denote the set of Reeb chords of length $0$ and $b\\leq - \\mu$ such that $H(x,r) = h(r) = \\mu r + b$ on $[1, \\infty) \\times \\partial M$.\n\\end{itemize}\nIf $H: \\widehat{M} \\to \\R$ is admissible and satisfies $H(x,r) = \\mu r + b$ on $[1, \\infty) \\times \\partial M$ we say that $H$ is admissible with \\textit{slope} $\\mu$ \\textit{(at infinity)}.\n \n\nDefine the action functional $\\mathcal{A}^{L_0 \\to L_1}_H = \\mathcal{A}_H : \\mathcal{P}_{L_0 \\to L_1} \\rightarrow \\R$ by \n\\[\n\\mathcal{A}_H(\\gamma) = f_0(x(0)) - f_1(x(1)) + \\int_{0}^{1} \\gamma^{*}\\lambda - \\int_{0}^{1} H(\\gamma(t))dt, \n\\]\nwhere $f_0$ and $f_1$ are functions on $L_0$ and $L_1$ respectively with $df_i = \\lambda|_{\\widehat{L}_i}, i= 0,1$. \nThe critical points of $\\mathcal{A}_H$ are Hamiltonian chords from $\\widehat{L}_0$ to $\\widehat{L}_1$ that reach $\\widehat{L}_1$ at time $1$. We define\n\\[\n \\mathcal{T}_{L_0\\to L_1}(H):=\\crit{\\mathcal{A}_H} = \\{\\gamma \\in \\mathcal{P}_{L_0 \\to L_1} \\mid \\dot{\\gamma}(t) = X_H(\\gamma(t))\\}, \n\\]\nand write $\\mathcal{T}_{L}(H)$ instead of $\\mathcal{T}_{L \\to L}(H)$. \nHere $X_H$ is the Hamiltonian vector field defined by $\\omega(X_H, \\cdot ) = -dH$. \nWe call an admissible Hamiltonian \\textit{non-degenerate} for $L_0 \\to L_1$ if \nall elements in $\\mathcal{T}_{L_0\\to L_1}(H)$ are non-degenerate, i.e. $\\phi^1_{X_H}(\\widehat{L_0})$ is transverse to $\\widehat{L_1}$. Such a Hamiltonian must have slope $\\mu \\notin \\mathcal{S}(M,L_0 \\to L_1)$. Note that every admissible Hamiltonian can be made non-degenerate for $L_0 \\to L_1$ after a generic perturbation (\\cite[Lemma 8.1]{AbouzaidSeidel2010}). We denote by \n \\begin{equation}\n \\mathcal{H}_{\\mathrm{reg}}(M,L_0 \\to L_1)\n\\end{equation}\nthe set of admissible Hamiltonians which are non-degenerate for $L_0 \\to L_1$.\nFor a Hamiltonian $H\\in \\mathcal{H}_{\\mathrm{reg}}(M,L_0 \\to L_1) $ all elements in $\\mathcal{T}_{L_0 \\to L_1}(H)$ have their image contained in $M$. \n\n\n For admissible Hamiltonians $H$ with slope $\\mu \\notin \\mathcal{S}$ that are constant in $M$ away from the boundary, depend on $r$ and increase sharply near $\\partial M$, $\\mathcal{T}_{L_0\\to L_1}(H)$ corresponds to $\\mathcal{T}^{\\mu}_{\\Lambda_0 \\to \\Lambda_1}(\\alpha_M)$ and intersection points of $L_0$ and $L_1$ in $M$. If $(M,L_0 \\to L_1)$ is regular, such Hamiltonians belong to the set $ \\mathcal{H}_{\\mathrm{reg}}(M,L_0 \\to L_1)$. \n\n If $(\\alpha_M, \\Lambda_0 \\to \\Lambda_1)$ is regular but $(M,L_0 \\to L_1)$ is not, we can take $H \\in \\mathcal{H}_{\\mathrm{reg}}(M,L_0 \\to L_1)$ to be a $C^2$-small negative function away from the boundary of $M$, and to depend only on $r$ and increase sharply near $\\partial M$. Then, $\\mathcal{T}_{L_0\\to L_1}(H)$ will correspond to $\\mathcal{T}^{\\mu}_{\\Lambda_0 \\to \\Lambda_1}(\\alpha_M)$ and intersection points of $L_0$ and $L_1$ in $M$ that are not destroyed by the Hamiltonian flow of $H$. \n\n\nAn almost complex structure $J$ on $\\left((0,\\infty) \\times \\partial M, \\lambda = r\\alpha_M \\right)$ is called \\textit{cylindrical} if it preserves $\\xi_{M}= \\ker \\alpha_M$, if $J|_{\\xi_{M}}$ is independent of $r$ and compatible with $d(r\\alpha_M)|_{\\xi_{M}}$, and if $J X_{\\alpha_M}=r\\partial_{r} $. \nIn the following we take almost complex structures $J$ on $\\widehat{M}$ that are \\textit{asymptotically cylindrical}, i.e. cylindrical on $[r, \\infty) \\times \\partial M$ for some $r>1$. \nThe $L^2$-gradient of the action functional with respect to the Riemannian metric given by $d\\lambda(J \\cdot,\\cdot) = g(\\cdot, \\cdot)$\nis given by\n\\[\n\\nabla \\mathcal{A}_H(\\gamma) = -J(\\gamma)\\left(\\partial_t \\gamma - X_H(\\gamma) \\right), \n\\]\nand we interpret the negative gradient flow lines as Floer strips\n\\begin{equation}\n\\begin{split}\\label{Floer}\n&u:\\R \\times [0,1] \\rightarrow \\widehat{M}, \\\\\n&\\overline{\\partial}_{J,H}(u)=\\partial_s u + J(u)(\\partial_t u - X_H(u) ) = 0, \\\\\n&u(\\cdot,0) \\in \\widehat{L}_0, \\mbox{ and } u( \\cdot ,1) \\in \\widehat{L}_1. \n\\end{split}\n\\end{equation}\nWe define the moduli space of parametrized Floer strips connecting two critical points $x$ and $y$ of $\\mathcal{A}_H$\n\\begin{equation}\n\\begin{split}\n\\widetilde{\\mathcal{M}}(x,y, H, J) = \\{ u: \\R \\times [0,1] \\rightarrow \\widehat{M} \\, | \\, u \\text{ satisfies \\eqref{Floer} }, \\lim_{s\\rightarrow -\\infty} = x \\mbox{ and } \\lim_{s\\rightarrow +\\infty} = y \\}. \n\\end{split}\n\\end{equation}\nThere is a natural $\\R$-action on $\\mathcal{M}(x,y, H, J)$ coming from the translations in the domain.\nLetting $\\widetilde{\\mathcal{M}}^1(x,y, H, J) $ be the set of elements of $\\widetilde{\\mathcal{M}}(x,y, H, J) $ that have Fredholm index 1 we write\n\\begin{equation}\n\\begin{split}\n& \\mathcal{M}^0(x,y, H, J) := \\widetilde{\\mathcal{M}}^1(x,y, H, J) \/ \\R. \n\\end{split}\n\\end{equation}\nwhere the quotient is taken with respect to the $\\R$-action mentioned above.The \\textit{energy} of an element $u$ is\n\\[\nE(u) := \\int_{-\\infty}^{\\infty}|\\nabla \\mathcal{A}_H|_{L^2}^2 \\, ds \n= \\mathcal{A}_H(x)-\\mathcal{A}_H(y).\n\\]\n\nFor a generic $J$ and non-degenerate admissible $H$ define the wrapped Floer chain complex\n\\[\n\\mathrm{CW}(H,L_0 \\to L_1) = \\bigoplus_{x \\in \\crit(\\mathcal{A}_H)} \\Z_2 \\cdot x,\n\\]\nwith differential $\\partial : \\mathrm{CW}(H,L_0 \\to L_1) \\rightarrow \\mathrm{CW}(H,L_0 \\to L_1)$ given by \n\\[\n\\partial (x) = \\sum_{y \\in \\crit(\\mathcal{A}_H)} \\#_{\\Z_2} \\mathcal{M}^0(x,y,H,J) \\cdot y.\n\\]\nFor generic $J$ the differential is well-defined and moreover $\\partial^2= 0$. \nFor simplicity we will write $\\mathrm{CW}(H)$ instead of $\\mathrm{CW}(H,L_0 \\to L_1)$ when there is no possibility of confusion.\nIn this paper we are not concerned with gradings in $\\mathrm{CW}$.\nThe homology of $(\\mathrm{CW}(H,L_0 \\to L_1),\\partial)$ is called the wrapped Floer homology of $(H,L_0 \\to L_1)$ and is denoted by $\\mathrm{HW}(H;L_0 \\to L_1)$, or in short $\\mathrm{HW}(H)$.\n\nNext we consider continuation maps. \nLet $H_-$ and $H_+$ be non-degenerate admissible Hamiltonians with $H_+(x) \\geq H_-(x)$ for all $x \\in \\widehat{M}$, in short $H_+ \\succ H_-$. Take an increasing homotopy through admissible Hamiltonians $(H_s)_{s\\in \\R}$, $\\partial_s H_s \\geq 0$, with $H_s = H_{\\pm}$ near $\\pm \\infty$.\nFor elements in $\\mathcal{M}^0(x_-,x_+,H_s,J)$, i.e. Floer strips \n\\begin{equation}\n\\begin{split}\\label{contin}\n&u:\\R \\times [0,1] \\rightarrow \\widehat{M}, \\\\\n&\\overline{\\partial}_{J,H_s}(u) := \\partial_s u + J(\\partial_t u - X_{H_{s}}(u)) = 0, \\\\\n&\\lim_{s\\rightarrow \\pm \\infty} u(s,t) = x_{\\pm}, \\\\\n&u(\\cdot, 0) \\in \\widehat{L}_0, \\mbox{ and } u(\\cdot, 1) \\in \\widehat{L}_1, \n\\end{split}\n\\end{equation}\nwith Fredholm index 0 connecting $x_- \\in \\crit(\\mathcal{A}_{H_-})$ and $x_+ \\in \\crit(\\mathcal{A}_{H_+})$, the action difference \nis \n\\[\\mathcal{A}_{H_-}(x_-) -\\mathcal{A}_{H_+}(x_+) = E(u) + \\int_{\\R \\times [0,1]}\\partial_s H_s(u). \n\\]\nHence the action decreases under the continuation maps\n\\[ \n\\iota^{H_-,H_+} : \\mathrm{CW}(H_-) \\rightarrow \\mathrm{CW}(H_+),\n\\]\ngiven by \n\\[\n\\iota^{H_-,H_+} (x_-) = \\sum_{x^+ \\crit(\\mathcal{A}_{H_+})} \\#_{\\Z_2} \\mathcal{M}^0(x_-,x_+,H_s,J) \\cdot x_+. \n\\]\nDefine the wrapped Floer homology $\\mathrm{HW}(M,L_0 \\to L_1) := \\dlim_H\\mathrm{HW}(H;L_0 \\to L_1)$, where the direct limit is taken over all $H\\in \\mathcal{H}_{\\mathrm{reg}}(M,L_0 \\to L_1) $. \n\n\\begin{defn}\nThe homology $\\mathrm{HW}(M,L_0 \\to L_1)$ is the direct limit of the filtered directed system $\\widetilde{\\mathrm{HW}}(M,L_0 \\to L_1) = \\left(\\mathrm{HW}^a(M,L_0 \\to L_1)\\right)_{a \\in (0,\\infty)}$. Here \n\\[\\mathrm{HW}^a(M,L_0 \\to L_1) := \\dlim_{H} {\\mathrm{HW}^{a}(H;L_0 \\to L_1)},\n\\]\nwhere $\\mathrm{HW}^{a}(H;L_0 \\to L_1) $ is the homology of the Floer chain complex restricted to critical points of action less than $a$. The persistence maps $\\iota_{a \\rightarrow b}: {\\mathrm{HW}^a(M,L_0 \\to L_1)} \\to {\\mathrm{HW}^b(M,L_0 \\to L_1)}$ are induced by the natural maps $\\mathrm{HW}^a(H,L_0 \\to L_1) \\rightarrow \\mathrm{HW}^b(H,L_0 \\to L_1)$ that come from inclusions. We write $\\iota_{a}: {\\mathrm{HW}^a(M,L_0 \\to L_1)} \\to {\\mathrm{HW}(M,L_0 \\to L_1)}$ for the induced map from $\\mathrm{HW}^a(M,L_0 \\to L_1)$ to the direct limit ${\\mathrm{HW}(M,L_0 \\to L_1)}$. \n\\end{defn}\n\n\\color{black}\nLet $H'\\geq H$ be Hamiltonians in $ \\mathcal{H}_{\\mathrm{reg}}(M,L_0 \\to L_1) $ and $b\\geq a$. Let $ \\iota^{H,H'}_{a \\to b}: \\mathrm{HW}^a(H, L_0 \\to L_1) \\to \\mathrm{HW}^b(H', L_0 \\to L_1)$ by the continuation map induced by any non-decreasing homotopy from $H$ to $H'$. In case $b=+\\infty$ we write $ \\iota^{H,H'}_{a}: \\mathrm{HW}^a(H, L_0 \\to L_1) \\to \\mathrm{HW}(H', L_0 \\to L_1)$.\n\n\\begin{rem} \\label{rem:stationary}\nNotice that $ \\iota^{H,H}_{a \\to b}$ is the map induced by the chain level inclusion $\\mathrm{CW}^a(H,L_0 \\to L_1) \\hookrightarrow \\mathrm{CW}^b(H,L_0 \\to L_1)$. For this reason we will also denote this inclusion also by $\\iota^{H,H}_{a \\to b}: \\mathrm{CW}^a(H,L_0 \\to L_1) \\hookrightarrow \\mathrm{CW}^b(H,L_0 \\to L_1)$.\n\\end{rem}\n\nBy the construction of $\\widetilde{\\mathrm{HW}}(M,L_0 \\to L_1)$ presented above we have for every number $a\\geq 0$ and Hamiltonian $H\\in \\mathcal{H}_{\\mathrm{reg}}(M,L_0 \\to L_1) $ a map \n\\begin{equation}\n\\chi^H_{a \\to a}:\\mathrm{HW}^a(H, L_0 \\to L_1) \\to \\mathrm{HW}^a(M, L_0 \\to L_1).\n\\end{equation}\nThis allows us to define for every Hamiltonian $H\\in \\mathcal{H}_{\\mathrm{reg}}(M,L_0 \\to L_1) $ and numbers $b\\geq a$ the map \n\\begin{equation*}\n\\chi^H_{a \\to b}:=\\iota_{a\\to b} \\circ \\chi^{H}_{a \\to a} = \\mathrm{HW}^a(H, L_0 \\to L_1) \\to \\mathrm{HW}^b(M, L_0 \\to L_1).\n\\end{equation*}\nUsing functoriality properties of continuation maps it is straightforward to check that $$ \\chi^H_{a \\to b} =\\chi^H_{b \\to b} \\circ \\iota^{H,H}_{a \\to b }. $$ For simplicity, in the case $b=+\\infty$ we write $$ \\chi^H_{a } = \\iota_{a} \\circ \\chi^{H}_{a \\to a} : \\mathrm{HW}^a(H, L_0 \\to L_1) \\to \\mathrm{HW}(M, L_0 \\to L_1).$$ \n \n \nFor each $H\\in \\mathcal{H}_{\\mathrm{reg}}(M,L_0 \\to L_1) $ we also have a map $$ \\chi^H: \\mathrm{HW}(H, L_0 \\to L_1) \\to \\mathrm{HW}(M, L_0 \\to L_1).$$ To define it, we first notice that since $ \\mathcal{T}_{L_0 \\to L_1}(H)$ is a finite set we can choose a number $ a_H>\\max_{x \\in \\mathcal{T}_{L_0 \\to L_1}(H)}\\{\\mathcal{A}(x)\\}$. For this choice of $a_H$ the chain complexes $ (\\mathrm{CW}(H, L_0 \\to L_1),\\partial) $ and $ ( \\mathrm{CW}^{a_H}(H, L_0 \\to L_1),\\partial)$ are identical, and we get $ \\mathrm{HW}(H, L_0 \\to L_1) = \\mathrm{HW}^{a_H}(H, L_0 \\to L_1)$. We then define $\\chi^H := \\chi^H_{a_H}$. It is an elementary exercise to check that the definition of $\\chi^H $ does not depend on the choice of $a_H>\\max_{x \\in \\mathcal{T}_{L_0 \\to L_1}(H)}\\{\\mathcal{A}(x)\\}$. In the same way we can construct for each $b >\\max_{x \\in \\mathcal{T}_{L_0 \\to L_1}(H)}\\{\\mathcal{A}(x)\\}$ a map $$ \\chi^H_{\\to b}: \\mathrm{HW}(H, L_0 \\to L_1) \\to \\mathrm{HW}^b(M, L_0 \\to L_1).$$\n\n\nThese maps are useful for the study of spectral numbers done in the next section. We will need the identity \n\\begin{equation}\\label{eq:naturality'}\n \\chi^H_a = \\chi^{H'} \\circ \\iota^{H,H'}_a,\n \\end{equation}\nwhich is established in an elementary way from the functoriality properties of continuation maps. In particular, we have\n\\begin{equation}\\label{eq:lepolepo}\n \\chi^H = \\chi^{H'} \\circ \\iota^{H,H'},\n \\end{equation}\n and\n\\begin{equation}\\label{eq:naturality}\n \\chi^H_a = \\chi^{H} \\circ \\iota^{H,H}_a.\n \\end{equation}\n\n\n\n\n\n\n\n\n\n \n We will now define the symplectic growth rate of $\\mathrm{HW}$.\n\\begin{defn} \\label{defi:Gamma_symp}\nThe exponential symplectic growth rate $\\Gamma^{\\mathrm{symp}}(M,L_0 \\to L_1)$ is defined by \n\\begin{equation}\n\\Gamma^{\\mathrm{symp}}(M,L_0 \\to L_1) := \\limsup_{a \\to \\infty} \\frac{\\log (\\dim \\mathrm{Im} \\ \\iota_a)}{a} = \\widetilde{\\Gamma}(\\widetilde{\\mathrm{HW}}(M,L_0 \\to L_1)).\n\\end{equation}\nAnalogously, given a family $(L_i)_{i\\in I}$ of asymptotically conical exact Lagrangians in $M$ we define $\\Gamma_{i\\in I}^{\\mathrm{symp}}(M,L_0 \\to L_{i}) := \\widetilde{\\Gamma}_{i\\in I}(\\widetilde{\\mathrm{HW}}(M,L_0 \\to L_i)_{i\\in I})$, where $\\widetilde{\\Gamma}_{i\\in I}(\\widetilde{\\mathrm{HW}}(M,L_0 \\to L_i)_{i\\in I})$ is defined as in Definition \\ref{defi:fam_growth}.\n\\end{defn}\n\n\\subsubsection{Spectral numbers in $\\mathrm{HW}$}\n\n\\begin{defn}\nAs ${\\mathrm{HW}(M,L_0 \\to L_1)}$ is the direct limit of the the f.d.s. $\\widetilde{\\mathrm{HW}}(M,L_0 \\to L_1) $, we define the spectral number $c$ of elements of ${\\mathrm{HW}(M,L_0 \\to L_1)}$ via the recipe given in Section~\\ref{subsec:algebra}.\n\\end{defn}\n\nWe now present an equivalent definition of $c$ which is more geometrical. Given $H\\in \\mathcal{H}_{\\mathrm{reg}}(M,L_0 \\to L_1) $, and a cycle $w \\in \\mathrm{CW}(H, L_0 \\to L_1)$ we denote by $[w] \\in \\mathrm{HW}(H, L_0 \\to L_1)$ the homology class of $w$ in $\\mathrm{HW}(H, L_0 \\to L_1)$. The cycle $w$ can be expressed in a unique way as a sum of elements of $\\mathcal{T}_{L_0 \\to L_1}(H)$ and we denote by $\\mathcal{A}(w)$ the maximum of the actions of these elements. \n\n If $w' \\in \\mathrm{CW}^a(H, L_0 \\to L_1)$, then it can be expressed in a unique way as a sum of elements in $\\mathcal{T}^a_{L_0 \\to L_1}(H)$. This expression is identical to the one of $\\iota_a^{H,H}(w')$, from what we conclude\n\\begin{equation}\n\\mathcal{A}(\\iota_a^{H,H}(w')) < a \\mbox{ for all } w' \\in \\mathrm{CW}^a(H, L_0 \\to L_1).\n\\end{equation}\n The right hand side in the following identity is often taken as the definition of the spectral number~$c(\\mathfrak{h})$.\n\\begin{lem} \\label{lem:minimax}\nFor a homology class $\\mathfrak{h}\\in\\mathrm{HW}(M, L_0 \\to L_1)$ we have\n\\begin{equation} \\label{eq:minmax}\nc(\\mathfrak{h}) = \\inf_{H\\in \\mathcal{H}_{\\mathrm{reg}}(M,L_0 \\to L_1)} \\{\\mathcal{A}(w) \\mid w\\in \\mathrm{CW}(H, L_0 \\to L_1) \\mbox{ is a cycle with } \\chi^H ([w]) = \\mathfrak{h} \\}.\n\\end{equation}\n\\end{lem}\n\\textit{Proof:} Let $H\\in \\mathcal{H}_{\\mathrm{reg}}(M,L_0 \\to L_1) $ and $w\\in \\mathrm{CW}(H, L_0 \\to L_1)$ be a cycle with $\\chi^H ([w]) = \\mathfrak{h}$. For each $a > \\mathcal{A}(w) $ we know that there exists a cycle $w' \\in \\mathrm{CW}^a(H, L_0 \\to L_1)$ such that $\\iota_a^{H,H}(w')=w$. Using \\eqref{eq:naturality} we obtain \n\\begin{equation*}\n \\chi^{H}_a([w'])= \\chi^{H}\\circ\\iota^{H,H}_{a}( [w']) = \\chi^{H} ( [w]) = \\mathfrak{h}.\n\\end{equation*}\nThis implies that $\\mathfrak{h}$ is in the image of $\\chi^{H}_a$ and thus in the image of $\\iota_a$, from what we get $c(\\mathfrak{h})\\leq a$. Since this is valid for each $a > \\mathcal{A}(w) $ we obtain that $c(\\mathfrak{h}) \\leq \\mathcal{A}(w)$, and it follows that\n\\begin{equation} \\label{eq:minmaxll}\nc(\\mathfrak{h}) \\leq \\inf_{H\\in \\mathcal{H}_{\\mathrm{reg}}(M,L_0 \\to L_1)} \\{\\mathcal{A}(w) \\mid w\\in \\mathrm{CW}(H, L_0 \\to L_1) \\mbox{ is a cycle with } \\chi^H ([w]) = \\mathfrak{h} \\}.\n\\end{equation}\n\nTo obtain the reverse inequality let $a > c(\\mathfrak{h})$. Then there exists $\\beta \\in \\mathrm{HW}^a(M, L_0 \\to L_1)$ such that $\\iota_a(\\beta) = \\mathfrak{h}$. By the construction of $\\mathrm{HW}^a(M, L_0 \\to L_1)$ we know that there exists $H \\in \\mathcal{H}_{\\mathrm{reg}}(M,L_0 \\to L_1)$ and a cycle $w' \\in \\mathrm{CW}^a(H, L_0 \\to L_1)$ such that $\\chi^H_{a \\to a} ([w'])=\\beta$. It follows that $$ \\chi^H_a ([w'])=\\iota_a \\circ \\chi^H_{a \\to a} ([w']) = \\iota_a(\\beta)=\\mathfrak{h}.$$\nLet $w:= \\iota^{H,H}_{a}( w')$. By the observation we made before the lemma we have $\\mathcal{A}(w) < a$. Using \\eqref{eq:naturality} we obtain $$ \\chi^H([w])= \\chi^H(\\iota^{H,H}_a([w'])= \\chi^H_a([w'])= \\mathfrak{h}. $$\n\nWe have shown that for each $a > c(\\mathfrak{h})$ there exists $H \\in \\mathcal{H}_{\\mathrm{reg}}(M,L_0 \\to L_1)$ and a cycle $w \\in \\mathrm{CW}(H, L_0 \\to L_1)$ such that $\\mathcal{A}(w) < a$ and $\\chi^H([w]) = \\mathfrak{h}$. It follows that \n\\begin{equation}\nc(\\mathfrak{h}) \\geq\\inf_{H\\in \\mathcal{H}_{\\mathrm{reg}}(M,L_0 \\to L_1)} \\{\\mathcal{A}(w) \\mid w\\in \\mathrm{CW}(H, L_0 \\to L_1) \\mbox{ is a cycle with } \\chi^H ([w]) = \\mathfrak{h} \\}.\n\\end{equation} \\qed\n\n\\subsubsection{A special type of Hamiltonians} \\label{rem:rem1}\n\n\n\n\nGiven an admissible Hamiltonian $H$ in $M$ and a number $a> 0$ we write $H\\prec a$ if the slope of $H$ is $ \\mathsf{K}(M,L_0 \\to L_1)$ a careful choice of a cofinal family of Hamiltonians shows that $\\mathrm{HW}^a(M,L_0 \\to L_1)$ is isomorphic to $\\dlim_{H\\prec a}\\mathrm{HW}(H; L_0 \\to L_1)$, where the direct limit is taken only over all non-degenerate admissible $H$ with slope less than $a$.\n\nTo explain this we first take a collar neighbourhood $\\mathfrak{V}=([1-\\delta,1] \\times \\Sigma) \\subset M$ of $\\partial M$ on which $L_0$ and $L_1$ are conical, and $\\lambda$ is given by $r\\alpha_M$. Since $a> \\mathsf{K}(M,L_0 \\to L_1)$ we can choose $ \\mathsf{K}(M,L_0 \\to L_1)<\\mu \\mathsf{K}(M,L_0 \\to L_1)$ we have \n\\begin{equation} \\label{eq:coisabonita}\n\\mathrm{HW}^a(M,L_0 \\to L_1)\\cong \\dlim_{H\\prec a}\\mathrm{HW}(H; L_0 \\to L_1)\n\\end{equation}\n\n\n\n\\color{black}\n\n\n\n\\subsection{Algebra and module structures on wrapped Floer homology} \\label{sec:productonHW}\n\n\\subsubsection{Algebra structure in $\\mathrm{HW}$}\nLet $L$ be an exact asymptotically conical Lagrangian on a Liouville domain $M$. We endow $M$ with an asymptotically cylindrical almost complex structure as in Section \\ref{subsec:def}. We recall the definition of the triangle product in the wrapped Floer homology $\\mathrm{HW}(M,L)$, and follow the conventions of \\cite{AS-product}. \n\nWe first define the triangle $\\Delta$. One first takes the disjoint union $\\mathbb{R} \\times [-1,0] \\cup \\mathbb{R} \\times [0,1]$. We identify the points $(s,0^-) \\in \\mathbb{R} \\times [-1,0] $ and $(s,0^+) \\in \\mathbb{R} \\times [0,1] $ for all $s \\geq 0$, and denote the resulting space by $\\Delta$. Let $p_{\\mathrm{sing}}$ be the point in $\\Delta$ which comes from the points $(0,0^-) \\in \\mathbb{R} \\times [-1,0]$ and $(0,0^+) \\in \\mathbb{R} \\times [0,1] $. \n\nThe interior of $\\Delta$ coincides with $(\\mathbb{R} \\times (-1,1)) \\setminus ((-\\infty,0] \\times \\{0\\})$. As $(\\mathbb{R} \\times (-1,1)) \\setminus ((-\\infty,0] \\times \\{0\\})$ is a subset of $\\mathbb{C}$ we can restrict the complex structure of $\\mathbb{C}$ to $(\\mathbb{R} \\times (-1,1)) \\setminus ((-\\infty,0] \\times \\{0\\})$. We then obtain a complex structure $j$ in the interior of $\\Delta$. This extends to a complex structure on $\\Delta\\setminus p_{\\mathrm{sing}}$. \nUsing again that the interior of $\\Delta$ coincides with $(\\mathbb{R} \\times (-1,1)) \\setminus ((-\\infty,0] \\times \\{0\\})$, we can define global coordinates $(s,t)$ on $\\Delta \\setminus p_{\\mathrm{sing}}$.\n\nFor an admissible Hamiltonian $H$ on $\\widehat{M}$, the solutions of the Floer equation on $\\Delta$ are maps $u: \\Delta \\to \\widehat{M}$ that satisfy\n\\begin{equation}\n\\overline{\\partial}_{J,H}(u):= \\partial_s u + J(u)(\\partial_t u - X_H(t,u))=0.\n\\end{equation}\n We write $\\widehat{H} =2H \\in C^{\\infty}(M)$.\n\n\n\nGiven $x_1, x_2\\in \\mathcal{T}_{L}(H)$ and $y \\in \\mathcal{T}_{L}(\\widehat{H})$ we let \n$\\mathcal{M}(x_1,x_2;y,L,J)$ be the space of maps $u: \\Delta \\to \\widehat{M}$ that satisfy $\\overline{\\partial}_{J,H}(u)=0$ and such that \n$u(z) \\in L$ for all $z\\in \\partial(\\Delta)$, $\\lim_{s\\to -\\infty} u(s,t-1)=x_1(t)$ for $t \\in [0,1]$, $\\lim_{s\\to -\\infty} u(s,t)=x_2(t)$ for $t \\in [0,1]$, and $\\lim_{s\\to +\\infty} u(s,2t-1)=y(t)$ for $t \\in [0,1]$. Define $n(x_1,x_2;y)$ as the number of elements of $\\mathcal{M}(x_1,x_2;y,L,J)$ which have Fredholm index $0$. If the moduli spaces $\\mathcal{M}(x_1,x_2;y,L,J)$ are transversely cut out, something that can be achieved by perturbing $H$ and $J$, the numbers $n(x_1,x_2;y)$ are always finite.\n\n\n Define $\\Upsilon_L: \\mathrm{CW}(H,L) \\otimes \\mathrm{CW}(H,L) \\to \\mathrm{CW}(\\widehat{H},L) $ by\n\\begin{equation}\n\\Upsilon_L(x_1,x_2)= \\sum_{y \\in \\mathcal{T}_L(\\widehat{H})}(n(x_1,x_2;y)\\mod 2) y\n\\end{equation}\nfor $x_1,x_2\\in \\mathcal{T}_{L}(H)$, and extending it linearly to $ \\mathrm{CW}(H,L) \\otimes \\mathrm{CW}(H,L)$.\nIt is proved in \\cite{AS-product} that $\\Upsilon_L$ descends to a map $H \\Upsilon_L: \\mathrm{HW}(H,L) \\otimes \\mathrm{HW}(H,L) \\to \\mathrm{HW}(\\widehat{H},L)$, that endows $ \\mathrm{HW}(H,L)$ with a product which we denote by $\\star$. \n It is compatible with the continuation maps, as follows by the results in \\cite[Chapter 5]{Schwarz-thesis}, and passing to the direct limit $H\\Upsilon_L$ endows $ \\mathrm{HW}(M,L)$ with a product. For homology classes $\\mathfrak{h},\\mathfrak{h}' \\in \\mathrm{HW}(M,L) $ we will also denote their product by $\\mathfrak{h} \\star \\mathfrak{h}'$. \nThe product $\\star$ is associative: the proof is identical to the proof in \\cite{Schwarz-thesis} that the pair of pants product in Floer homology is associative.\nAs $\\star$ is distributive with respect to the vector space structure of $ \\mathrm{HW}(M,L)$ it gives $ \\mathrm{HW}(M,L)$ the structure of a ring. Since we defined $\\mathrm{HW}(M,L)$ with coefficients in $\\mathbb{Z}_2$ the product $\\star$ actually endows $\\mathrm{HW}(M,L)$ with the structure of an algebra.\n\n\n\nIt was proved in \\cite{AS-product} that in the case where $M =T^*Q$ of a compact manifold $Q$ and $L=T_q Q$ for some point $q\\in Q$, the triangle product coincides with the Pontrjagin product.\n\n\\color{black}\nAn important property of the triangle product is given by \n\\begin{lem} \\label{lem:subad}\nThe spectral numbers $c$ of ${\\mathrm{HW}}(M,L)$ are subadditive with respect to $\\star$.\n\\end{lem}\n \\textit{Proof:} We will need the triangle inequality \n\\begin{equation}\\label{triangle_inequ}\n\\mathcal{A}_{\\widehat{H}}(y) \\leq \\mathcal{A}_{H}(x_1) + \\mathcal{A}_{H}(x_2),\n\\end{equation}\nthat must be satisfied by the actions of $x_1,x_2 \\in \\mathcal{T}_{L}(H)$ and $y \\in \\mathcal{T}_{L}(\\widehat{H})$ if the moduli space $\\mathcal{M}(x_1,x_2;y,L,J) \\neq \\emptyset$ (see \\cite[Formula 3.18]{AS-product}).\n\nLet $\\mathfrak{h}_1,\\mathfrak{h}_2 \\in {\\mathrm{HW}}(M,L)$. Given $\\delta>0$, we know from Lemma \\ref{lem:minimax} that there exits Hamiltonians $H_1, H_2 \\in \\mathcal{H}_{\\mathrm{reg}}(M,L)$ and cycles $w'_i \\in \\mathrm{CW}(H_i,L)$ such that\n\\begin{eqnarray*}\n \\chi^{H_i}([w'_i]) = \\mathfrak{h}_i \\ \\mbox{ and } \\ \\mathcal{A}(w'_i)< c(\\mathfrak{h}_i) +\\frac{\\delta}{2}\n\\end{eqnarray*}\n for $i=1,2$.\n Let now $H\\in \\mathcal{H}_{\\mathrm{reg}}(M,L)$ such that $H\\geq H_1$ and $H\\geq H_2$. We define $w_i:= \\iota^{H_i,H}(w'_i)$ for $i=1,2$. Since the action decreases under the continuation maps $\\iota^{H_i,H}$ we have $\\mathcal{A}(w_i)< c(\\mathfrak{h}_i) + \\frac{\\delta}{2}$, and using \\eqref{eq:lepolepo} we obtain\n \\begin{equation*}\n \\chi^H([w_i])=\\chi^{H} ( \\iota^{H_i,H}([w'_i])) = \\chi^{H_i}([w'_i])=\\mathfrak{h}_i, \n \\end{equation*}\n for $i=1,2$.\n By \\eqref{triangle_inequ} we have $\\mathcal{A}(\\Upsilon_L(w_1 \\otimes w_2))\\leq c(\\mathfrak{h}_1) +c(\\mathfrak{h}_2) +\\delta$. By definition $[\\Upsilon_L(w_1 \\otimes w_2)]= [w_1]\\star[w_2]$, and\n by our construction of $\\star$ in $\\mathrm{HW}(M,L)$ we have \n \\begin{equation*}\n \\chi^{\\widehat{H}}([w_1]\\star[w_2]) = \\chi^H([w_1])\\star \\chi^H([w_2]) = \\mathfrak{h}_1 \\star \\mathfrak{h}_2.\n \\end{equation*}\n It then follows from Lemma \\ref{lem:minimax} that $c( \\mathfrak{h}_1 \\star \\mathfrak{h}_2) \\leq \\mathcal{A}(\\Upsilon_L(w_1 \\otimes w_2)) \\leq c(\\mathfrak{h}_1) +c(\\mathfrak{h}_2) +\\delta$. \n \n Summing up, we have shown that for any $\\delta>0$ we have $c( \\mathfrak{h}_1 \\star \\mathfrak{h}_2) K = \\mathsf{K}(M,L_0 \\to L_1)$, defined in \\eqref{K(M,L)}, homomorphisms \n\\[\nj_{!}(L_0,L_1)_a : \\mathrm{HW}^a(M,L_0 \\to L_1) \\rightarrow \\mathrm{HW}^a(W, L^{'}_{0} \\to L^{'}_{1})\n\t\\]\nthat are compatible with the persistence morphisms $\\iota_{a\\rightarrow b}$, for $K< a < b$. Moreover, the homomorphisms are functorial with respect to a composition of embeddings $W_1 \\subset W_2 \\subset M$ and the induced maps in the direct limit \n\\[\n\\bar{j}_{!}(L_0) = \\bar{j}_{!}(L_0,L_0): \\mathrm{HW}(M,L_0) \\rightarrow \\mathrm{HW}(W,L^{'}_0), \\text{ and} \n\\] \n\\[\n\\bar{j}_{!}(L_0,L_1): \\mathrm{HW}(M,L_0 \\to L_1) \\rightarrow \\mathrm{HW}(W, L^{'}_0 \\to L^{'}_1)\n\\] \nare compatible with the algebra and module structure, i.e.\n\\begin{equation}\\label{vitproduct}\n\\bar{j}_{!}(L_0) (x \\star y) = \\bar{j}_{!}(L_0)(x) \\star \\bar{j}_{!}(L_0)(y)\n\\end{equation}\nand\n\\begin{equation}\\label{vitmodule}\n \\bar{j}_{!}(L_0,L_1)(x \\ast z) = \\bar{j}_{!}(L_0)(x) \\ast \\bar{j}_{!}(L_0,L_1)(z)\n\\end{equation}\nfor all $x, y \\in \\mathrm{HW}(M,L_0)$ and $z \\in \\mathrm{HW}(M,L_0, L_1)$.\n\nWe first give the definition of ${j}_{!}(L_0,L_1)$. We may assume that $(M,L_0 \\to L_1)$ and $(W,L^{'}_0 \\to L^{'}_1)$ are regular. \\color{black} Otherwise we can perform the construction considering suitable compactly supported Hamiltonian perturbations of $L_0$ and $L_1$. \\color{black} Let $\\mathcal{S} := \\mathcal{S}(M,L_0 \\to L_1) \\cup \\mathcal{S}(W,L^{'}_0 \\to L^{'}_1)$. We furthermore assume that actually $W \\subset M_{\\tau^2}$ for some $\\tau < 1$, sufficiently close to $1$. One can get the maps for general $W \\subset M$ by an inverse limit. \n\nFirst of all, for every $R > 1$ one can construct a compactly supported Hamiltonian isotopy $(\\psi^R_t)_{t\\in [0,1]}$ on $\\widehat{M}$, ($\\psi^R_0 = \\id$, $\\psi:= \\psi^R_1$) that leaves $\\widehat{L}_0$ invariant and maps $\\widehat{L}_1$ to a Lagrangian $\\widehat{L}^R_1$ that is conical on $(\\widehat{M} \\setminus M_R) \\cup (W_R \\setminus W)$ and that is transfer admissible for the pair $(M_R, W_R)$ as follows. Map $L_1 \\setminus W$ by the Liouville flow $(\\phi_{\\log t})_{t\\in [1,R]}$ into $A_R = M_R \\setminus W_R$. Since $L_1$ is conical near $\\partial W$, we can extend $\\left(L^{'}_1 \\cup \\phi_{\\log{t}}(L_1 \\setminus W)\\right)_{t\\in [1,R]}$ to a 1-parameter family of exact Lagrangians interpolating between $\\widehat{L}_1$ and a Lagrangian $\\widehat{L}^R_1$. Therefore we can choose a Hamiltonian isotopy $(\\psi^R_t)_{t \\in [0,1]}$ in $\\widehat{M}$ that realizes this Lagrangian isotopy and is supported in $M_{\\frac{1}{\\tau}R} \\setminus W_{\\tau}$. Since $\\widehat{L}_0$ is conical outside $W$, we can choose the isotopy to leave $\\widehat{L}_0$ invariant. We can choose the isotopy such that $(\\psi\\circ\\zeta)^{*}\\lambda = R\\zeta^{*}\\lambda$,\nwhere $\\zeta: {L}_1 \\setminus W \\hookrightarrow \\widehat{M}$ is the embedding of $L_1$ restricted to $L_1 \\setminus W$. The function\n\\[\nf_R: \\widehat{L}^R_1 \\to \\R, \\text{ with } f_R(x) = \\begin{cases} f_1(x), &\\text {if } x\\in L_1 = \\widehat{L}^R_1\\cap W, \\\\\n Rf_1(\\psi^{-1}x), &\\text{ elsewhere}\n\t\t\t\\end{cases}\n\\]\nis a primitive of $\\lambda|_{\\widehat{L}^R_1}$.\n \n\nWe now carefully choose for every $\\mu \\notin \\mathcal{S}$ sufficiently large a step-shaped Hamiltonian $H^{step}_{\\mu}$ on $\\widehat{M}$. \nLet $k_W := \\min\\{f_0(x) - f_1(x) \\, | \\, x \\in L_0 \\cap L_1 \\cap W \\}$ where $f_i$ are the primitives of $\\lambda|_{L_i}$, $i=0,1$. Let $\\widetilde{k} = \\max\\{-k_W, 0\\}$. Let \n\\[\n\\widetilde{K} = \\mathsf{K}(M,W,L_0 \\to L_1) = \\max\\{ \\max \\{f_0(x) - f_1(x) \\, | \\, x \\in L_0 \\cap L_1 \\cap M\\setminus W\\}, 0\\}.\n\\]\nChoose a small $\\epsilon > 0$. \nLet $\\mu > \\widetilde{K}$, $\\mu \\notin \\mathcal{S}$, and let $\\delta_{\\mu} = \\min \\{\\dist(\\mu,\\mathcal{S}), \\mu - \\widetilde{K} \\}$. Choose $R > \\frac{\\widetilde{k} + \\mu + 4\\epsilon}{\\delta_{\\mu}}$. \n\\input{figure1.tex}\nWe choose a smooth function $H^{step}_{\\mu}: \\widehat{M} \\rightarrow \\R$ that only depends on the radial coordinate $r= r_W$ in $(0,R) \\times \\partial W$ and only on the radial coordinate $r=r_M$ in $(\\tau R,\\infty) \\times \\partial M$, and such that\n\\begin{equation}\\label{step}\nH^{step}_{\\mu}(x) = \\begin{cases}\n-\\epsilon,&\\text{ if }x \\in W_{\\tau} \\\\\n\\frac{\\partial^2 H}{\\partial r} \\geq 0, &\\text{ if } x = (r,y) \\in W \\setminus W_{\\tau} \\\\\n\\mu r - \\mu, &\\text{ if } x=(r,y) \\in W_{\\tau R} \\setminus W \\\\\n\\frac{\\partial^2 H}{\\partial r} \\leq 0, &\\text{ if } x = (r,y) \\in W_R \\setminus W_{\\tau R} \\\\\n(R-1)\\mu - \\epsilon, &\\text{ if } x \\in M_{\\tau R} \\setminus W_R \\\\\n\\frac{\\partial^2 H}{\\partial r} \\geq 0, &\\text{ if } x = (r,y) \\in M_R \\setminus M_{\\tau R} \\\\\n\\mu r -\\mu, &\\text{ if }x = (r,y) \\in \\widehat{M} \\setminus M_R.\n\\end{cases}\n\\end{equation}\n\nWe divide the critical points of the action functional $\\mathcal{A} := \\mathcal{A}_{H_{\\mu}^{step}}^{\\widehat{L}_0 \\to \\widehat{L}^R_1}$ of $H_{\\mu}^{step}$ with respect to $\\widehat{L}_0$ and $\\widehat{L}^R_0$ into four classes: Intersections of $L_0$ and $L_1$ in $W_{\\tau}$ denoted by $\\mathfrak{A}^{*}$, Hamiltonian chords close to $\\partial W$ denoted by $\\mathfrak{A}^{**}$, intersections of $\\widehat{L}_0$ and $\\widehat{L}^R_1$ in $M_R \\setminus W_R$ denoted by $\\mathfrak{B}^{*}$, and chords close to $\\partial W_R$ and $\\partial M_R$ denoted by $\\mathfrak{B}^{**}$. \nWe can estimate the action values as follows. \n\\begin{align}\n\\mathcal{A}(x) &\\geq k_w - \\epsilon \\geq -\\widetilde{k} - \\epsilon, \\text{ if } x \\in \\mathfrak{A}^{*}, \\\\\n\\mathcal{A}(x) &> -\\epsilon \\geq -\\widetilde{k} - \\epsilon, \\text{ if } x \\in \\mathfrak{A}^{**}, \\\\\n\\mathcal{A}(x) &\\leq R\\widetilde{K} - ((R-1)\\mu -\\epsilon) < - \\widetilde{k} - 3\\epsilon, \\text{ if } x \\in \\mathfrak{B}^{*}, \\text{ and} \\label{B*}\\\\\n\\mathcal{A}(x) &< (\\mu - \\dist(\\mu, \\mathcal{S}))R - ((R-1)\\mu -\\epsilon) < -\\widetilde{k} - 3\\epsilon, \\text{ if } x \\in \\mathfrak{B}^{**}.\n\\end{align}\n\nIn \\eqref{B*} we use that $f_0(x) - f_R(x) \\leq \\widetilde{K}R$ for every $x \\in \\mathfrak{B}^{*}$. \n\nAltogether we get that $\\mathcal{A}(x) \\geq -\\widetilde{k} - \\epsilon $, if $x\\in \\mathfrak{A} = \\mathfrak{A}^{*} \\cup \\mathfrak{A}^{**}$ and $\\mathcal{A}(x) < -\\widetilde{k} - 3\\epsilon$, if $x\\in \\mathfrak{B} = \\mathfrak{B}^{*} \\cup \\mathfrak{B}^{**}$. Hence there are no Floer trajectories from $\\mathfrak{B}$ to $\\mathfrak{A}$. So\n $\\mathrm{CW}^{(-\\widetilde{k} - 2\\epsilon, +\\infty)}_{*}(H^{step}_{\\mu}; \\widehat{L}_0 \\to \\widehat{L}^R_1) = \\mathrm{CW}_*(H^{step}_{\\mu})\/\\mathrm{CW}^{(-\\infty, -\\widetilde{k} - 2\\epsilon)}_{*}(H^{step}_{\\mu})$ generated by elements of action larger then $-\\widetilde{k} - 2\\epsilon$ is a chain complex, and the projection $\\mathrm{CW}(H^{step}_{\\mu}) \\rightarrow \\mathrm{CW}^{(-\\widetilde{k} - 2\\epsilon, +\\infty)}(H^{step}_{\\mu})$ induces a map \n\\begin{equation}\\label{vit}\n\\mathrm{HW}(H^{step}_{\\mu}; \\widehat{L}_0 \\to \\widehat{L}^R_1) \\rightarrow \\mathrm{HW}^{(-\\widetilde{k} - 2\\epsilon,+\\infty)}(H^{step}_{\\mu}; \\widehat{L}_0 \\to \\widehat{L}^R_1)\n\\end{equation}\non homology. \n\nLet now $H^M_{\\mu}$ be a non-degenerate admissible Hamiltonian with respect to $M$ on $\\widehat{M}$ with slope $\\mu$, and $H^W_{\\mu}$ a non-degenerate admissible Hamiltonian with respect to $W$ on $\\widehat{W}$ with slope $\\mu$. We have the isomorphisms \n\\begin{align}\n&\\mathrm{HW}(H_{\\mu}^M; L_0 \\to L_1) \\overset{\\cong}\\rightarrow \\mathrm{HW}((\\psi^{-1})^{*}H_{\\mu}^M; \\widehat{L}_0 \\rightarrow \\widehat{L}^R_1) \\overset{\\cong}\\rightarrow \\mathrm{HW}(H^{step}_{\\mu}; \\widehat{L}_0 \\to \\widehat{L}^R_1), \\text{ and} \\label{H^M}\\\\\n&\\mathrm{HW}^{(-\\widetilde{k} - 2\\epsilon,+\\infty)}(H^{step}_{\\mu}; \\widehat{L}_0 \\to \\widehat{L}^R_1) \\overset{\\cong}\\rightarrow \\mathrm{HW}(H_{\\mu}^W; L^{'}_0 \\to L^{'}_1). \\label{H^W} \n\\end{align}\n\nHere, the second isomorphism in \\eqref{H^M} holds, since $(\\psi^{-1})^{*}H_{\\mu}^M$ and $H^{step}_{\\mu}$ can be connected by a compactly supported homotopy of Hamiltonians. To get the isomorphism in $\\eqref{H^W}$ we choose a conical almost complex structure near $\\partial W$. By \\cite[Lemma 7.2]{AbouzaidSeidel2010}, see also \\cite[Appendix D]{Ritter2013} there are no Floer trajectories with asymptotics in $W$ that leave $W$ and hence the differential of $\\mathrm{CW}^{(-\\widetilde{k} - 2\\epsilon,+\\infty)}(H^{step}_{\\mu}, \\widehat{L}_0 \\to \\widehat{L}^R_1)$ only counts Floer trajectories that map into $W$.\n\nCombining \\eqref{vit}, \\eqref{H^M}, and \\eqref{H^W} gives maps\n\\begin{equation}\\label{vit1}\n j_{\\mu}: \\mathrm{HW}(H^M_{\\mu}; L_0 \\to L_1) \\rightarrow \\mathrm{HW}(H^W_{\\mu}; L^{'}_0 \\to L^{'}_1) \n\\end{equation}\nfor any $\\mu > \\widetilde{K}$, $\\mu \\notin \\mathcal{S}$. \nThe isomorphisms \\eqref{vit}, \\eqref{H^M}, and \\eqref{H^W} are all compatible with Floer continuation maps induced by monotone increasing homotopies of the corresponding Hamiltonians. We do not give the details here and refer the reader to \\cite[Theorem 9.8]{Ritter2013}. We thus get commutative diagrams\n\\[\n\\begin{CD}\n \\mathrm{HW}(H^M_{\\mu}; L_0 \\to L_1) @>{j_{\\mu}}>> \\mathrm{HW}(H^W_{\\mu}; L^{'}_0 \\to L^{'}_1) \\\\\n @V{\\iota^{H^M_{\\mu}, H^M_{\\eta}}}VV @V{\\iota^{H^W_{\\mu}, H^W_{\\eta}}}VV \\\\\n \\mathrm{HW}(H^M_{\\eta}; L_0 \\to L_1) @>{j_{\\eta}}>> \\mathrm{HW}(H^W_{\\eta}; L^{'}_0 \\to L^{'}_1)\n\\end{CD}\n\\]\nfor any $\\eta > \\mu > \\widetilde{K}$, $\\mu, \\eta \\notin \\mathcal{S}$. \n\nHence, for any $a > K=\\mathsf{K}(M,L_0 \\to L_1) \\geq \\widetilde{K}$ one obtains, because of the construction in Section \\ref{rem:rem1}, a map \n\\[\nj_{!}(L_0,L_1)_a : \\mathrm{HW}^a(M, L_0 \\to L_1) \\rightarrow \\mathrm{HW}^a(W, L^{'}_0 \\to L^{'}_1) \n\\]\ninduced in the direct limit taken over all non-degenerate admissible Hamiltonians with slope $\\mu$, $K <\\mu < a$. By the construction these maps are compatible with the persistence morphisms $\\iota_{a \\rightarrow b}$, for $K < a < b$. \n\n\nBy a standard compactness-cobordism argument, and by using once again the non-escaping result \\cite[Lemma 7.2]{AbouzaidSeidel2010} one can show the compatibility of the algebra and module structure with the Viterbo transfer maps \\eqref{vitproduct} and \\eqref{vitmodule}; for this see \\cite{Ritter2013}.\n\n\n\n\n\\subsection{Change of the contact hypersurface $\\partial M$}\n\nFrom the Viterbo transfer one can deduce invariance properties of $\\mathrm{HW}$ under a graphical change of $\\partial M$ in $\\widehat{M}$. This will be used to bound the growth rate of Reeb chords for different choices of contact forms on $(\\partial M, \\xi_{M})$. \nLet $M$ be a Liouville domain with asymptotically conical exact Lagrangians $L_0$ and $L_1$ as above, let $0 < \\epsilon < 1$.\n\\begin{lem}\\label{M_epsilon}\nAssume that $L_i$, $i=0,1$, are conical on $M\\setminus M_{\\epsilon}$. Then, for $a > K = \\mathsf{K}(M,L_0 \\to L_1)$, we have $\\mathrm{HW}^a(M_{\\epsilon},L_0 \\cap M_{\\epsilon} \\to L_1 \\cap M_{\\epsilon}) \\overset{\\varphi_a}{\\cong} \\mathrm{HW}^{\\frac{1}{\\epsilon}a}(M,L_0 \\to L_1)$.\nMoreover, the Viterbo map $\\mathrm{HW}^a(M, L_0 \\to L_1) \\rightarrow \\mathrm{HW}^a(M_{\\epsilon},L_0 \\cap M_{\\epsilon} \\to L_1 \\cap M_{\\epsilon})$ composed with $\\varphi_a$ is the persistence morphism $\\mathrm{HW}^a(M, L_0 \\to L_1) \\rightarrow \\mathrm{HW}^{\\frac{1}{\\epsilon}a}(M,L_0 \\to L_1)$. \n\\end{lem}\n\\textit{Proof: }\nNote, that adding a constant to any Hamiltonian $H$ or applying a compactly supported deformation to $H$ does not change its Floer homology. \nLet $H$ be an admissible Hamiltonian with slope $\\mu$ with respect to $M_{\\epsilon}$. Then $H-\\mu(\\frac{1}{\\epsilon}-1)$ is an admissible Hamiltonian with slope $\\frac{1}{\\epsilon} \\mu$ with respect to $M$. Moreover, if one chooses a cofinal sequence of Hamiltonians of the first kind with slopes $K< \\mu < a$, there are compactly supported homotopies of the shifted Hamiltonians to a cofinal sequence with respect to $M$ with slopes $\\frac{1}{\\epsilon}\\mu$. This gives the first statement. \n\nObserve, that both the Viterbo transfer map in the present situation and the persistence morphisms are given by a continuation map induced by a monotone homotopy. One can apply a usual chain homotopy argument in Floer homology to see the second statement.\n\\qed\n\nLet $\\mathsf{f}: \\partial M \\rightarrow [1, \\infty)$ be a smooth function. Recall that $M_{\\mathsf{f}} = \\widehat{M} \\setminus \\{(r,x) \\, | \\, r > \\mathsf{f}(x), x \\in \\partial M\\}$. Let $\\zeta = \\max_{\\partial M} \\mathsf{f}$. \n\\begin{lem}\\label{lem:changeofhypersurface}\nThe filtered directed systems $(\\mathrm{HW}^a(M,L_0 \\to L_1))_{a\\in (0,\\infty)}$ and $(\\mathrm{HW}^a(M_{\\mathsf{f}},\\widehat{L}_0 \\cap M_{\\mathsf{f}} \\to \\widehat{L}_0 \\cap M_{\\mathsf{f}} ))_{a\\in (0, \\infty)}$ are $(\\zeta, 1)$-interleaved. \n\\end{lem}\n\\textit{Proof: }\nThe morphisms of filtered directed systems $f$ and $g$, with\n$f_a: \\mathrm{HW}^a(M,L_0 \\to L_1) \\cong \\mathrm{HW}^{\\zeta a}(M_{\\zeta}, L_0 \\to L_1) \\to \\mathrm{HW}^ {\\zeta a}(M_{\\mathsf{f}},\\widehat{L}_0 \\cap M_{\\mathsf{f}}, \\widehat{L}_1 \\cap M_{\\mathsf{f}})$ and $g_a:\\mathrm{HW}^a(M_{\\mathsf{f}},\\widehat{L}_0 \\cap M_{\\mathsf{f}}, \\widehat{L}_1 \\cap M_{\\mathsf{f}}) \\to \\mathrm{HW}^a(M,L_0 \\to L_1)$, given by Viterbo maps, yield by functoriality of Viterbo maps and Lemma \\ref{M_epsilon} the $(\\zeta,1)$- interleaving of $(\\mathrm{HW}^a(M,L_0 \\to L_1))_{a\\in (0,\\infty)}$ and $(\\mathrm{HW}^a(M_{\\mathsf{f}},\\widehat{L}_0 \\cap M_{\\mathsf{f}} \\to \\widehat{L}_0 \\cap M_{\\mathsf{f}} ))_{a\\in (0, \\infty)}$.\n\\qed\n\n\n\\section{From algebraic growth to positivity of topological entropy}\\label{sec:entropy}\nIn this section we prove Theorem \\ref{theorementropy}.\n\n\n\n\\color{black}\n\\subsection{Legendrian isotopies, transfer admissible Lagrangians and growth} \\label{subsec:technical}\n\n\n\nWe start by introducing some notation. Let $M=(Y,\\omega,\\lambda)$ be a Liouville domain and $L$ be an asymptotically conical exact Lagrangian disk in $M$. We denote by $\\Lambda$ the Legendrian sphere $\\partial L$.\n Letting $\\Sigma := \\partial M$ and $\\alpha_{M} := \\lambda |_{\\Sigma}$ be the contact form induced by $M$ on $\\Sigma$ we assume that $(\\alpha_{M}, \\Lambda \\to \\Lambda)$ is regular. As usually, we denote by $\\xi_{M}$ the contact structure $\\ker \\alpha_{M}$.\n \n \n\nOur approach to prove invariance of the exponential symplectic growth of $\\mathrm{HW}$ differs from the ones developed by \\cite{MacariniSchlenk2011,McLean2012}. It makes extensive use of the module and algebra structures that exist on $\\mathrm{HW}$.\n We will need the following\n \\begin{defn}\nLet $\\mu>0$ and $\\Lambda_0$ be a Legendrian sphere in $(\\Sigma, \\xi_{M})$. Assume that $\\Lambda_1$ is Legendrian isotopic to $\\Lambda_0$. We say that $\\Lambda_1$ is \\textit{$\\mu$-close to $\\Lambda_0$ in the $C^3$-sense} if there exists a Legendrian isotopy $\\theta:[-1,1] \\times S^{n-1} \\to (\\Sigma, \\xi_{M})$ from $\\Lambda_0$ to $\\Lambda_1$ whose $C^3$-norm is $<\\mu$, and which is stationary in the first coordinate outside a compact subset of $(-1,1)$.\n\\end{defn}\n\nRecall that the symplectisation of a contact form $\\alpha$ on $(\\Sigma, \\xi_{M})$ is the exact symplectic manifold $((0,+\\infty)\\times \\Sigma, dr\\alpha, r\\alpha)$ where $r$ denotes the first coordinate in $(0,+\\infty)\\times \\Sigma$.\nThe following lemma is essentially due to Chantraine \\cite{Baptiste} and is proved in Appendix B.\n\\begin{lem} \\label{lemmaBaptiste}\nFix a constant $\\epsilon >0$, a contact form $\\alpha$ on $(\\Sigma,\\xi)$, a Legendrian $\\Lambda_0$ in $(\\Sigma,\\xi)$, and a tubular neighbourhood $U(\\Lambda_0)$ of $\\Lambda_0$ in $\\Sigma$. Then there exists $\\delta>0$ such that if $\\Lambda_1$ is $\\delta$-close to $\\Lambda_0$ in the $C^3$-sense, then there exist exact Lagrangian cobordisms $\\mathcal{L}^-$ from $\\Lambda_1$ to $\\Lambda_0$ and $\\mathcal{L}^+$ from $\\Lambda_0$ to $\\Lambda_1$ in the symplectization of $\\alpha$ satisfying:\n\\begin{itemize}\n\\item [a)] $\\mathcal{L}^- $ is conical outside $[1-\\frac{\\epsilon}{2}, 1-\\frac{\\epsilon}{4}] \\times \\Sigma$,\n\\item [b)] $\\mathcal{L}^+$ is conical outside $[1+\\frac{\\epsilon}{4}, 1+ \\frac{\\epsilon}{2}] \\times \\Sigma$,\n\\item [c)] the projections of $\\mathcal{L}^+$ and $\\mathcal{L}^-$ to $\\Sigma$ are completely contained in $U(\\Lambda_0)$,\n\\item [d)] the primitives $f^\\pm$ of $(r\\alpha) |_{\\mathcal{L^\\pm}}$ have support in $[1-\\frac{\\epsilon}{2}, 1-\\frac{\\epsilon}{4}] \\times \\Sigma$ and $[1+\\frac{\\epsilon}{4}, 1+ \\frac{\\epsilon}{2}] \\times \\Sigma$, respectively, and $| f^\\pm |_{C^0} < \\epsilon$.\n\\end{itemize}\nMoreover if $\\mathcal{L}$ is the exact Lagrangian cylinder obtained by gluing $\\mathcal{L}^+ \\cap [1,+\\infty) \\times \\Sigma)$ on top of $\\mathcal{L}^- \\cap ((0,1] \\times \\Sigma)$ we have that\n\\begin{itemize}\n\\item [e)] $\\mathcal{L}$ is Hamiltonian isotopic to $\\mathbb{R}\\times \\Lambda_0$ in the symplectization of $\\alpha$, and the Hamiltonian producing the isotopy can be taken to have support in $[1- \\frac{\\epsilon}{2},1+ \\frac{\\epsilon}{2}] \\times \\Sigma$.\n\\end{itemize}\n\\end{lem}\n \n We now fix $\\epsilon > 0$ such that $L$ is conical on $M \\setminus M_{1 - 2\\epsilon}$. We choose a Legendrian tubular neighbourhood $\\mathcal{U}(\\Lambda)$ of $\\Lambda$ on $(\\Sigma,\\xi_M) $. For these choices of $\\epsilon>0$ and $\\mathcal{U}(\\Lambda)$, we choose $\\delta_1>0$ given by Lemma \\ref{lemmaBaptiste}.\n \n We then choose a Legendrian sphere $\\Lambda_1$ which is $\\delta_1$-close to $\\Lambda$ in the $C^3$ sense, is disjoint from $\\Lambda$, and satisfies that $(\\alpha_{M}, \\Lambda \\to \\Lambda_1)$ is regular.\n\nIt follows from Lemma \\ref{lemmaBaptiste} that there exists an exact Lagrangian cobordism $\\mathcal{L}^-$ from $\\Lambda_1$ to $\\Lambda$ in the symplectization of $\\alpha_{M}$ which is conical\noutside $[1-\\frac{\\epsilon}{2},1-\\frac{\\epsilon}{4}] \\times \\Sigma$. We can then glue $\\mathcal{L}^- \\cap[1-\\frac{\\epsilon}{2},1] \\times \\Sigma$ to $L \\cap M_{1-\\frac{\\epsilon}{2}}$ to obtain an exact Lagrangian submanifold $L_1$ in $M$. The Lagrangian $L_1$ is an exact filling of $\\Lambda_1$. Let $f_L$ be the primitive of $\\lambda\\mid_{L}$ which vanishes in $\\Lambda$. Using Lemma \\ref{lemmaBaptiste} we can glue $f^-$ to the restriction of $f_L$ to $L \\cap M_{1-\\frac{\\epsilon}{2}}$ to obtain primitive of $f_{L_1}$ of $\\lambda\\mid_{L_1}$ which vanishes in $\\Lambda_1$. \n\nBecause of the control given by Lemma \\ref{lemmaBaptiste} on the function $| f^- |_{C^0}$ on $\\mathcal{L}^-$, and the facts that $L$ and $L_1$ coincide on $M_{1-\\frac{\\epsilon}{2}}$ and $f_L$ vanishes on $L \\cap(M\\setminus M_{1-\\frac{\\epsilon}{2}})$ we have\n\\begin{equation} \\label{eq:control:LtoL_1}\n\\mathsf{K}(M,L \\to L_1) < \\epsilon.\n\\end{equation}\n\nBy Lemma \\ref{lemmaBaptiste} d) the Lagrangian $L_1$ is transfer admissible for the pair $(M,M_{1-\\epsilon})$. Combining this with \\eqref{eq:control:LtoL_1} we obtain for each $a>\\epsilon\\geq \\mathsf{K}(M,L \\to L_1)$ a Viterbo map $\\Psi^a_{\\mathcal{L}^-} : \\mathrm{HW}^a(M,L \\to L_1) \\to \\mathrm{HW}^a(M_{1-\\epsilon},L)$, where to simplify notation we keep denoting by $L$ and $L_1$ the restrictions of $L$ and $L_1$ to $M_{1-\\epsilon}$. Passing to the direct limit we obtain a map $\\Psi_{\\mathcal{L}^-} : \\mathrm{HW}(M,L \\to L_1) \\to \\mathrm{HW}(M_{1-\\epsilon},L)$.\n\n\nBy Lemma \\ref{lemmaBaptiste} we also have an exact Lagrangian cobordism $\\mathcal{L}^+$ from $\\Lambda$ to $\\Lambda_1$, which is diffeomorphic to $\\mathbb{R}\\times S^{n-1}$, and is conical over $\\Lambda$ for $r\\geq 1 + \\frac{\\epsilon}{2}$ and conical over $\\Lambda_1$ for $r \\leq 1 + \\frac{\\epsilon}{4} $. By gluing $\\mathcal{L}^+ \\cap ([1, 1+ \\epsilon] \\times \\Sigma)$ to $L_1$ we obtain an exact Lagrangian $\\overline{L}$ in $M_{1+\\epsilon}$. By Lemma \\ref{lemmaBaptiste} d) the Lagrangian $\\overline{L}$ is transfer admissible for the pair $(M_{1+\\epsilon},M)$. By gluing $f^+$ to $f_{L_1}$ we obtain a primitive $f_{\\overline{L}}$ of $\\lambda\\mid_{\\overline{L}}$. Reasoning as in the proof of \\eqref{eq:control:LtoL_1} one obtains\n\\begin{equation} \\label{eq:control:LtoL_1above}\n\\mathsf{K}(M_{1+\\epsilon},L \\to \\overline{L}) < \\epsilon.\n\\end{equation}\n We thus obtain for each $a>\\epsilon$ a Viterbo map $\\Psi^a_{\\mathcal{L}^+} : \\mathrm{HW}^a(M_{1+\\epsilon},L \\to \\overline{L}) \\to \\mathrm{HW}^a(M,L \\to L_1 )$, where by abuse of notation we denote by $L$ the conical extension of $L$ to $M_{1+\\epsilon}$. Passing to the direct limit we obtain a map $\\Psi_{\\mathcal{L}^+} : \\mathrm{HW}(M_{1+\\epsilon},L \\to \\overline{L}) \\to \\mathrm{HW}(M,L \\to L_1 )$.\n\n\nBy Lemma \\ref{lemmaBaptiste}, $\\overline{L}$ is Hamiltonian isotopic to the conical extension of $L$ to $M_{1+\\epsilon}$, which we will still denote by $L$, for a Hamiltonian function which vanishes outside $M_{1+\\frac{\\epsilon}{2}} \\setminus M_{1- \\frac{\\epsilon}{2}}$. \nA continuation argument then implies that for each admissible Hamiltonian $H$ that is regular for both $(M_{1+\\epsilon},L \\to \\overline{L})$ and $(M_{1+\\epsilon}, L)$ and has slope $>\\epsilon$ we have that $\\mathrm{HW}(H,L \\to \\overline{L})$ and $\\mathrm{HW}(H, L)$ are isomorphic. By Section \\ref{rem:rem1} we conclude that for each $a>\\epsilon$ \n the wrapped Floer homologies \n\\begin{equation} \\label{eq:isocont}\n\\mathrm{HW}^a(M_{1+\\epsilon},L \\to \\overline{L}) \\mbox{ and } \\mathrm{HW}^a(M_{1+\\epsilon}, L) \\mbox{ are isomorphic}. \n\\end{equation}\nThis induces an isomorphism $\\Phi: \\mathrm{HW}(M_{1+\\epsilon}, L) \\to \\mathrm{HW}(M_{1+\\epsilon},L \\to \\overline{L})$.\n\n\n\nSince $L$ is conical on $M_{1+\\epsilon} \\setminus M_{1-\\epsilon}$, $M \\setminus M_{1-\\epsilon}$ and $M_{1+\\epsilon} \\setminus M$, we have transfer maps\n\\begin{itemize}\n\\item[] $\\Psi^{\\pm}_{L} : \\mathrm{HW}(M_{1+\\epsilon}, L) \\to \\mathrm{HW}(M_{1-\\epsilon}, L)$, \n\\item[] $\\Psi^{-}_{L} : \\mathrm{HW}(M, L) \\to \\mathrm{HW}(M_{1-\\epsilon}, L)$,\n\\item[] $\\Psi^{+}_{L} : \\mathrm{HW}(M_{1+\\epsilon}, L) \\to \\mathrm{HW}(M, L)$.\n \\end{itemize}\nWe notice that the contact forms induced by $\\lambda$ on $\\{1-\\epsilon\\} \\times \\Sigma$ and $\\{1+\\epsilon\\} \\times \\Sigma$ are \n$\\frac{\\alpha_{M}}{1-\\epsilon}$ and $\\frac{\\alpha_{M}}{1+\\epsilon}$, respectively. Thus, as explained in Lemma \\ref{M_epsilon}, the maps $\\Psi^\\pm_L$, $\\Psi^-_L$ and $\\Psi^+_L$\n are induced by asymptotic isomorphisms of f.d.s. For this reason we will denote by $A_L$ the algebras $ \\mathrm{HW}(M_{1+\\epsilon}, L)$, $\\mathrm{HW}(M_{1-\\epsilon}, L)$ and $ \\mathrm{HW}(M, L)$. More generally, the same reasoning shows that for any $\\zeta > -\\epsilon$ the algebra $\\mathrm{HW}(M_{1+\\zeta}, L)$ is isomorphic to $ \\mathrm{HW}(M_{1-\\epsilon}, L)$ by an asymptotic isomorphism. \n\nThe homologies $\\mathrm{HW}(M, L \\to L_1)$, $\\mathrm{HW}(M_{1+\\epsilon}, L \\to \\overline{L})$ and $\\mathrm{HW}(M_{1-\\epsilon}, L )$ are modules over the algebras $\\mathrm{HW}(M, L)$, $\\mathrm{HW}(M_{1+\\epsilon}, L)$ and $\\mathrm{HW}(M_{1-\\epsilon}, L)$, respectively: they are therefore $A_L$-modules.\nBy this discussion and \\eqref{vitmodule} in section \\ref{sec:Viterbo} the maps $\\Phi$, $\\Psi_{\\mathcal{L}^-}$ and $\\Psi_{\\mathcal{L}^+}$ are $A_L$-module homomorphisms.\n\nBy functoriality of continuation maps, the diagram \n\\\\\n\\[\n\\begin{CD}\n \\mathrm{HW}(M_{1+\\epsilon}, L\\to \\overline{L}) @<\\Phi<< \\mathrm{HW}(M_{1+\\epsilon},L ) \\\\\n@V{\\Psi_{\\mathcal{L}^-}\\circ\\Psi_{\\mathcal{L}^+}}VV @V\\Psi^{\\pm}_{L}VV \\\\\n \\mathrm{HW}(M_{1-\\epsilon}, L ) @<\\mathrm{id}<<\\mathrm{HW}(M_{1-\\epsilon}, L) \\\\\n\\end{CD}\n\\]\n\\\\\nis commutative. It thus follows that the map ${\\Psi_{\\mathcal{L}^-}\\circ\\Psi_{\\mathcal{L}^+}}$ is an $A_L$-module isomorphism. We thus conclude that $\\Psi_{\\mathcal{L}^+}$ is injective. \nLet $\\mathbf{1}_L$ be the unit in $\\mathrm{HW}(M_{1+\\epsilon},L )$. As $\\Phi$ is an $A_L$-module isomorphism and $\\Psi_{\\mathcal{L}^+}$ is an injective $A_L$-module homomorphism we know that the element $m_{L_1} :=\\Psi_{\\mathcal{L}^+} \\circ \\Phi( \\mathbf{1}_L)$ in $\\mathrm{HW}(M,L \\to L_1 )$ is a stretching element.\nWe have thus proved the following:\n\\begin{lem} \\label{lemmaprelim} \nThe wrapped Floer homology $\\mathrm{HW}(M,L \\to L_1 )$ is a stretched module over $\\mathrm{HW}(M,L)$. It follows from Lemma \\ref{mod_growth}, Lemma \\ref{lem:subad}, and Lemma \\ref{lem:subad2} that \n\\begin{equation} \n\\Gamma^{\\mathrm{symp}}(M,L \\to L_1 ) \\geq {\\Gamma^{\\mathrm{symp}}(M,L )}. \n\\end{equation} \n\\end{lem}\n\nRecall that our Legendrian sphere $\\Lambda_1$ was chosen disjoint from $\\Lambda$. It follows that intersections of the Lagrangian disk $L_1$ and $L$ are contained in $M_{1-\\frac{\\epsilon}{5}}$.\nBy a small Hamiltonian isotopy supported inside $M_{1-\\frac{\\epsilon}{5}}$ we can perturb $L_1$ to an exact Lagrangian $L'_1$ that is transverse to $L$. We take the perturbation to be small enough so that there is a primitive $f_{L'_1}$ of $\\lambda\\mid_{L'_1}$ which vanishes in $\\partial L_1'$ and satisfies \n\\begin{equation} \\label{eq:control:LtoL'_1}\n\\mathsf{K}(M_{1},L \\to L'_1) < \\epsilon.\n\\end{equation}\nA continuation argument identical to the one used in the proof of \\eqref{eq:isocont} implies that for $a>\\epsilon$ \n the homologies $\\mathrm{HW}^a(M, L \\to L_1)$ and $\\mathrm{HW}^a(M, L \\to L'_1)$ are isomorphic. \n\nWe let \n\\begin{equation}\n\\mathsf{C}_{\\mathrm{regium}}:= \\#( L'_1 \\cap L).\n\\end{equation}\n This number will be useful later for estimates of the growth of the number of Reeb chords.\n\n\nWe now consider a tubular neighbourhood $\\widetilde{\\mathcal{U}}(\\Lambda_1) $ which does not intersect $\\Lambda$. By Lemma \\ref{lemmaBaptiste} there exists $\\delta_2>0$ such that if a Legendrian sphere $\\Lambda_2$ is $\\delta_2$-close to $\\Lambda_1$ in the $C^3$-sense, then there exist exact Lagrangian cobordisms $\\mathcal{L}_{2 \\to 1}$ from $\\Lambda_2$ to $\\Lambda_1$, and $\\mathcal{L}_{1 \\to 2}$, from $\\Lambda_1$ to $\\Lambda_2$, both contained in the symplectization of $\\alpha_{M}$.\n It follows from Lemma \\ref{lemmaBaptiste} that by taking $\\delta_2>0$ smaller, if necessary, we can guarantee that\n\\begin{itemize}\n\\item $\\mathcal{L}_{2 \\to 1}$ is conical outside $[1-\\frac{\\epsilon}{5}, 1-\\frac{\\epsilon}{6}] \\times \\Sigma$,\n\\item $\\mathcal{L}_{1 \\to 2}$ is conical outside $[1+\\frac{\\epsilon}{6}, 1+\\frac{\\epsilon}{5}] \\times \\Sigma$,\n\\item the projections of $\\mathcal{L}_{2 \\to 1}$ and $\\mathcal{L}_{1 \\to 2}$ to $\\Sigma$ are contained in $\\widetilde{\\mathcal{U}}(\\Lambda_1) $,\n\\item there exist primitives $f_{2 \\to 1}$ and $f_{1 \\to 2}$ of $r\\alpha_{M} |_{\\mathcal{L}_{2 \\to 1}}$ and $r\\alpha_{M} |_{\\mathcal{L}_{1 \\to 2}}$, respectively, with support in $[1-\\frac{\\epsilon}{5}, 1-\\frac{\\epsilon}{6}] \\times \\Sigma$ and $[1+\\frac{\\epsilon}{6}, 1+\\frac{\\epsilon}{5}] \\times \\Sigma$, respectively, such that $|f_{2 \\to 1}|_{C^0}<\\epsilon$ and $|f_{1 \\to 2}|<\\epsilon$,\n\\item the exact Lagrangian $\\mathcal{L}_{1 \\to 1}$ in the symplectisation of $\\alpha_{M}$ obtained by gluing $\\mathcal{L}_{1 \\to 2} \\cap ({[1,+\\infty)\\times \\Sigma})$ on top of $\\mathcal{L}_{2 \\to 1} \\cap ((0,1]\\times \\Sigma)$ is Hamiltonian isotopic to $(0,+\\infty)\\times \\Lambda_1$ for an isotopy which is stationary outside ${(1-\\frac{\\epsilon}{5},1+\\frac{\\epsilon}{5})\\times \\Sigma}$.\n\\end{itemize}\n It is clear that one can glue $f_{1 \\to 2}$ and $f_{2 \\to 1}$ to obtain a primitive $f_{1 \\to 1}$ of $r\\alpha_{M} |_{\\mathcal{L}_{1 \\to 1}}$ which satisfies $|f_{1 \\to 1}| < \\epsilon$.\n\nWe then glue $\\mathcal{L}_{2 \\to 1} \\cap ([1-\\frac{\\epsilon}{5}, 1] \\times \\Sigma)$ on top of $L_1 \\subset M_{1-\\frac{\\epsilon}{5}}$ to obtain an asymptotically conical exact Lagrangian $L_2$ with $L_2\\cap \\partial M= \\Lambda_2$. Let $L'_2$ be the exact Lagrangian submanifold obtained from gluing $\\mathcal{L}_{2 \\to 1} \\cap ([1-\\frac{\\epsilon}{5}, 1] \\times \\Sigma)$ on top of $L'_1 \\subset M_{1-\\frac{\\epsilon}{5}}$. It is clear that $L'_2$ and $L_2$ are Hamiltonian isotopic for a Hamiltonian which has support contained in $M_{1 - \\frac{\\epsilon}{6}}$. \n\n\n Notice that the intersection points of $L'_2 \\cap L$ are the same as the intersection points of $ L'_1 \\cap L$. We thus conclude:\n\\begin{equation}\n \\#(L'_2 \\cap L) = \\mathsf{C}_{\\mathrm{regium}}.\n\\end{equation}\nWe can glue $f_{2 \\to 1}$ to the restriction of $f_{L_1}$ to $L_1 \\cap M_{1-\\frac{\\epsilon}{5}}$ to obtain a primitive $f_{L_2}$ of $\\lambda\\mid_{L_2}$ such that \n\\begin{equation} \\label{eq:controlLtoL_2}\n\\mathsf{K}(M,L \\to L_2) < \\epsilon.\n\\end{equation}\nSimilarly, one obtains a primitive $f_{L'_2}$ of $\\lambda\\mid_{L'_2}$ such that \n\\begin{equation} \\label{eq:controlLtoL'_2}\n\\mathsf{K}(M,L \\to L'_2) < \\epsilon.\n\\end{equation}\nAssuming that $(\\alpha_{M}, \\Lambda \\to \\Lambda_2)$ is regular the Lagrangian $L_2$ is admissible for the pair $(M,M_{1-\\frac{\\epsilon}{5}})$.\nWe then obtain for each $a>\\epsilon$ a transfer map $\\Psi^a_{\\mathcal{L}_{2\\to 1}}: \\mathrm{HW}^a(M, L \\to L_2) \\to \\mathrm{HW}^a(M_{1-\\frac{\\epsilon}{5}}, L \\to L_1)$. These induce a map $ \\Psi_{\\mathcal{L}_{2\\to 1}}: \\mathrm{HW}(M, L \\to L_2) \\to \\mathrm{HW}(M_{1-\\frac{\\epsilon}{5}}, L \\to L_1)$.\n\nBy \\eqref{eq:controlLtoL_2} and \\eqref{eq:controlLtoL'_2} and the fact that $L_2$ and $L'_2$ are Hamiltonian isotopic for an isotopy supported inside $M_{1-\\frac{\\epsilon}{5}}$, we can apply the reasoning used to prove \\eqref{eq:isocont} to show that for each $a>\\epsilon$ \n\\begin{equation} \\label{equsefulestimate}\n \\mathrm{HW}^a(M, L \\to L_2) \\mbox{ and } \\mathrm{HW}^a(M, L \\to L'_2) \\mbox{ are isomorphic.}\n\\end{equation}\n\nGluing $\\mathcal{L}_{1 \\to 2} \\cap ([1,1+\\frac{\\epsilon}{5}] \\times \\Sigma)$ on top of $L_2 \\subset M$ we obtain an asymptotically conical Lagrangian $\\widetilde{L}_1$ in $M_{1+\\frac{\\epsilon}{5}}$ which is transfer admissible for the pair $(M_{1+\\frac{\\epsilon}{5}},M)$. Reasoning as in the proof of \\eqref{eq:control:LtoL_1} we obtain that \n\\begin{equation}\n\\mathsf{K}(M_{1+\\frac{\\epsilon}{5}},L \\to \\widetilde{L}_1) < \\epsilon.\n\\end{equation}\n We thus get for each $a>\\epsilon$ a transfer map $\\Psi^a_{\\mathcal{L}_{1\\to 2}}: \\mathrm{HW}^a(M_{1+\\frac{\\epsilon}{5}}, L \\to \\widetilde{L}_1) \\to \\mathrm{HW}^a(M, L \\to L_2)$, and in the direct limit a homomorphism $\\Psi_{\\mathcal{L}_{1\\to 2}}: \\mathrm{HW}(M_{1+\\frac{\\epsilon}{5}}, L \\to \\widetilde{L}_1) \\to \\mathrm{HW}(M, L \\to L_2)$.\n \n\n \n\nWe finally glue $\\mathcal{L}^+ \\cap ([1 +\\frac{\\epsilon}{5} ,1 + {\\epsilon}]\\times \\Sigma)$ on top of $\\widetilde{L}_1$ to obtain an asymptotically conical exact Lagrangian $\\widetilde{L}$ on $M_{1+\\epsilon}$. The Lagrangian $\\widetilde{L}$ is an exact filling of $\\Lambda$. It is clear from Lemma \\ref{lemmaBaptiste} that $\\widetilde{L}$ is Hamiltonian isotopic to $L$, for a Hamiltonian which has support contained in $M_{1+\\frac{\\epsilon}{2}} \\setminus M_{1-\\frac{\\epsilon}{2}}$. Reasoning as in the proof of \\eqref{eq:control:LtoL_1} we obtain a primitive $f_{\\widetilde{L}}$ of $\\lambda \\mid_{\\widetilde{L}}$ such that \n\\begin{equation} \\label{eq:controlLtoLhat}\n\\mathsf{K}(M_{1+\\epsilon},L \\to \\widetilde{L}) < \\epsilon.\n\\end{equation}\nWe claim that for every $a>\\epsilon$ there exists an isomorphism\n\\begin{equation}\\label{eq:isocrucial}\n \\Psi^a_{L, \\widetilde{L}}: \\mathrm{HW}^a(M_{1+\\epsilon}, L ) \\to \\mathrm{HW}^a(M_{1+\\epsilon}, L \\to \\widetilde{L}).\n \\end{equation}\n To establish this claim we first notice that if $H$ is a Hamiltonian in\\footnote{By \\cite[Lemma 8.1]{AbouzaidSeidel2010} any admissible Hamiltonian in $M_{1+\\epsilon}$ can be perturbed to one in $\\mathcal{H}_{\\mathrm{reg}}(M_{1+\\epsilon},L\\to \\widetilde{L}) \\cap \\mathcal{H}_{\\mathrm{reg}}(M_{1+\\epsilon},L)$.} $\\mathcal{H}_{\\mathrm{reg}}(M_{1+\\epsilon},L\\to \\widetilde{L}) \\cap \\mathcal{H}_{\\mathrm{reg}}(M_{1+\\epsilon},L)$ it follows from the fact that $\\widetilde{L}$ is Hamiltonian isotopic to $L$ for a Hamiltonian which has support contained in $M_{1+\\frac{\\epsilon}{2}} \\setminus M_{1-\\frac{\\epsilon}{2}}$ that there exists a continuation isomorphism $\\Psi_{H,L, \\widetilde{L}}: \\mathrm{HW}(H, L ) \\to \\mathrm{HW}(H, L \\to \\widetilde{L})$. Equation \\eqref{eq:isocrucial} then follows from combining these isomorphisms and the identifications $\\mathrm{HW}^a(M_{1+\\epsilon}, L ) \\cong \\dlim_{H\\prec a}\\mathrm{HW}(H; L)$ and $\\mathrm{HW}^a(M_{1+\\epsilon}, L ) \\cong \\dlim_{H\\prec a}\\mathrm{HW}(H; L \\to \\widetilde{L})$ for $a>\\epsilon\\geq \\max\\{\\mathsf{K}(M_{1+\\epsilon},L \\to \\widetilde{L}); \\mathsf{K}(M_{1+\\epsilon},L)\\}$ which were established in \\eqref{eq:coisabonita}. The maps $\\Psi^a_{L, \\widetilde{L}}$ are compatible with the persistence morphisms of the f.d.s. $\\widetilde{\\mathrm{HW}}(M_{1+\\epsilon}, L )$ and $\\widetilde{\\mathrm{HW}}(M_{1+\\epsilon}, L \\to \\widetilde{L})$ and induce an asymptotic morphism between them. On the direct limit we get a map\n \\begin{equation}\\label{eq:asymp}\n \\Psi_{L, \\widetilde{L}}: \\mathrm{HW}(M_{1+\\epsilon}, L ) \\to \\mathrm{HW}(M_{1+\\epsilon}, L \\to \\widetilde{L}).\n \\end{equation}\n \n\n \n The succession of exact Lagrangian submanifolds we constructed is schematically presented in Figure 2.\n \n \\input{figure3.tex} \\label{figure3}\n \n\n Since $\\widetilde{L}$ is transfer admissible for $(M_{1+\\epsilon}, M_{1+\\frac{\\epsilon}{5}})$ we also obtain for each $a>\\epsilon\\geq \\mathsf{K}(M_{1+\\epsilon},L \\to \\widetilde{L})$ a transfer map $\\Phi^a_{\\mathcal{L}^+} : \\mathrm{HW}^a(M_{1+\\epsilon}, L \\to \\widetilde{L}) \\to \\mathrm{HW}^a(M_{1+\\frac{\\epsilon}{5}}, L \\to \\widetilde{L}_1) $. This induces a homomorphism $\\Phi_{\\mathcal{L}^+} : \\mathrm{HW}(M_{1+\\epsilon}, L \\to \\widetilde{L}) \\to \\mathrm{HW}(M_{1+\\frac{\\epsilon}{5}}, L \\to \\widetilde{L}_1) $.\n \n Analogously, it follows from Lemma \\ref{lemmaBaptiste} that $L_1$ is transfer admissible for the pair $(M_{1-\\frac{\\epsilon}{5}}, M_{1-\\epsilon})$, which gives us for each $a>\\epsilon\\geq \\mathsf{K}(M,L \\to L_1) \\geq \\mathsf{K}(M_{1-\\frac{\\epsilon}{5}},L \\to L_1)$ a map $\\Phi^a_{\\mathcal{L}^-} : \\mathrm{HW}^a(M_{1-\\frac{\\epsilon}{5}}, L \\to {L}_1) \\to \\mathrm{HW}^a(M_{1-\\epsilon}, L)$. These homomorphisms induce a homomorphism $\\Phi_{\\mathcal{L}^-} : \\mathrm{HW}(M_{1-\\frac{\\epsilon}{5}}, L \\to {L}_1) \\to \\mathrm{HW}(M_{1-\\epsilon}, L)$.\n\n The following lemma will be important for the study of the growth rate of $\\mathrm{HW}(M, L \\to L_2)$.\n\n\\begin{lem} \\label{lem:cacareco}\nFor $0<\\delta_1$ and $0<\\delta_2$ chosen as above we have that the spectral number of $\\Psi_{L, \\widetilde{L}}( \\mathbf{1}_L)$ is $\\leq \\epsilon$.\n\\end{lem}\n\\textit{Proof:} \nWe know from \\cite{Ritter2013} that $ c(\\mathbf{1}_L) = 0$. This implies that for every $a\\geq 0$ the element $\\mathbf{1}_L $ is in the image of $\\iota_a: \\mathrm{HW}^a(M_{1+\\epsilon},L) \\to \\mathrm{HW}(M_{1+\\epsilon},L)$. \n\nLet $a>\\epsilon$. As remarked above, the maps $\\Psi^a_{L, \\widetilde{L}}$ are compatible with the persistence morphisms of $\\widetilde{\\mathrm{HW}}(M_{1+\\epsilon}, L )$ and $\\widetilde{\\mathrm{HW}}(M_{1+\\epsilon}, L \\to \\widetilde{L})$, which implies that the diagram\n\\\\\n\\[\n\\begin{CD}\n \\mathrm{HW}^a(M_{1+\\epsilon}, L ) @>\\Psi^a_{L, \\widetilde{L}}>> \\mathrm{HW}^a(M_{1+\\epsilon}, L \\to \\widetilde{L}) \\\\\n @V\\iota_aVV @V\\iota_aVV \\\\ \n \\mathrm{HW}(M_{1+\\epsilon}, L) @>\\Psi_{L, \\widetilde{L}}>>\\mathrm{HW}(M_{1+\\epsilon}, L \\to \\widetilde{L})\n\\end{CD}\n\\]\n\\\\\nis commutative. It follows that $\\Psi_{L, \\widetilde{L}}(\\mathbf{1}_L)$ is in the image of $\\iota_a: \\mathrm{HW}^a(M_{1+\\epsilon}, L \\to \\widetilde{L}) \\to \\mathrm{HW}(M_{1+\\epsilon}, L \\to \\widetilde{L})$, from what we obtain that $c(\\Psi_{L, \\widetilde{L}}(\\mathbf{1}_L)) \\leq a$. Since this is true for every $a>\\epsilon$ we conclude that $c(\\Psi_{L, \\widetilde{L}}(\\mathbf{1}_L)) \\leq \\epsilon$. \\qed\n\n\n\n\n\n\n\n\n\n\nBy our discussion so far we have transfer maps\n\\\\\n\\[\n\\begin{CD}\n\\mathrm{HW}(M_{1+\\epsilon}, L \\to \\widetilde{L}) @>\\Phi_{\\mathcal{L}^+}>> \\mathrm{HW}(M_{1+\\frac{\\epsilon}{5}}, L \\to \\widetilde{L}_1) \n@>\\Psi_{\\mathcal{L}_{1 \\to 2}}>> \\mathrm{HW}(M, L \\to {L}_2) \\\\ @. @. @V\\Psi_{\\mathcal{L}_{2\\to 1}}VV \\\\ @. \\mathrm{HW}(M_{1-\\epsilon}, L) @<\\Phi_{\\mathcal{L}^-}<< \\mathrm{HW}(M_{1-\\frac{\\epsilon}{5}}, L \\to L_1)\n\\end{CD}\n\\]\n\\\\\n\n Using the fact that $\\widetilde{L}$ is Hamiltonian isotopic to $L$ by a Hamiltonian with support contained in $M_{1+\\frac{\\epsilon}{2}} \\setminus M_{1-\\frac{\\epsilon}{2}}$ and reasoning identically as in the proof of Lemma \\ref{lemmaprelim} we conclude that the composition $\\Phi_{\\mathcal{L}^-} \\circ \\Psi_{\\mathcal{L}_{2\\to 1}} \\circ \\Psi_{\\mathcal{L}_{1 \\to 2}} \\circ \\Phi_{\\mathcal{L}^+} \\circ \\Psi_{L, \\widetilde{L}}$ is induced by an asymptotic isomorphism from $\\mathrm{HW}(M_{1+\\epsilon}, L )$ to $\\mathrm{HW}(M_{1-\\epsilon}, L )$.\nIt follows that $\\Psi_{\\mathcal{L}_{1 \\to 2}} \\circ \\Phi_{\\mathcal{L}^+} : \\mathrm{HW}(M_{1+\\epsilon}, L \\to \\widetilde{L}) \\to \\mathrm{HW}(M, L \\to {L}_2)$ is an injective $A_L$-module homomorphism. \nWe define $m_{L_2}:= \\Psi_{\\mathcal{L}_{1 \\to 2}} \\circ \\Phi_{\\mathcal{L}^+}(\\Psi_{L, \\widetilde{L}}( \\mathbf{1}_L))$.\nThe element $m_{L_2} \\in \\mathrm{HW}(M, L \\to {L}_2)$ is stretching since it is the image of a stretching element by an injective $A_L$-module homomorphism.\n\n\n\n By the behaviour of spectral numbers under transfer maps, combined with \\eqref{eq:controlLtoLhat} and Lemma \\ref{lem:cacareco} we conclude that \n\\begin{equation} \\label{uniformitymodules}\nc(m_{L_2}) \\leq \\max\\{c(\\Psi_{L, \\widetilde{L}}( \\mathbf{1}_L)),\\mathsf{K}(M_{1+\\epsilon}, L \\to \\widetilde{L}) \\} \\leq \\epsilon.\n\\end{equation}\n \n\nWe denote by $\\mathcal{V}_{\\alpha_{M}}(\\Lambda_1)$ the set of Legendrian spheres $\\Lambda_2$ in the same Legendrian isotopy class of $\\Lambda_1$ that are $\\delta_2$-close to $\\Lambda_1$ is the $C^3$-sense.\nLet $\\mathcal{V}_{\\alpha_{M}}^{ \\mathrm{reg}}(\\Lambda_1) \\subset \\mathcal{V}_{\\alpha_{M}}(\\Lambda_1)$ be the subset of these $\\Lambda_2$ for which, in addition, $(\\alpha_{M}, \\Lambda \\to \\Lambda_2)$ is regular. We denote by $L_2$ the filling of $\\Lambda_2$ constructed above.\n Our discussion so far implies the following\n\\begin{prop} \\label{uniformmodules2} \nThe family $(\\mathrm{HW}(M,L \\to L_2))_{\\Lambda_2 \\in \\mathcal{V}_{\\alpha_{M}}^{\\mathrm{reg}}(\\Lambda_1)}$ of $A_L$-modules is uniformly stretched. It follows from Lemma \\ref{mod_fam_growth}, Lemma \\ref{lem:subad}, and Lemma \\ref{lem:subad2} that \n\\begin{equation}\n\\Gamma_{\\Lambda_2 \\in \\mathcal{V}_{\\alpha_{M}}^{\\mathrm{reg}}(\\Lambda_1)}^{\\mathrm{symp}}(M,L \\to L_2) \\geq \\Gamma^{\\mathrm{symp}}(M, L).\n\\end{equation} \n\n\\end{prop}\n\\textit{Proof:} \nThe proposition follows directly from the fact that the element $m_{L_2} \\in \\mathrm{HW}(M,L \\to L_2)$ is stretching and from \\eqref{uniformitymodules}. \\qed\n\nLet $\\alpha$ be a contact form on $(\\Sigma,\\xi_{M})$. We assume that the function $\\mathsf{f}_\\alpha$ defined by $\\alpha= \\mathsf{f}_\\alpha \\alpha_{M}$ satisfies $\\mathsf{f}_\\alpha \\geq 1$. We thus have the inclusions $ M_{\\mathsf{f}_\\alpha} \\subset M_{\\max \\mathsf{f}_\\alpha}$ and $M \\subset M_{\\mathsf{f}_\\alpha}$.\n\nWe denote by $\\mathcal{V}_{\\alpha_{M}}^{ \\alpha-\\mathrm{reg}}(\\Lambda_1) \\subset \\mathcal{V}_{\\alpha_{M}}^{ \\mathrm{reg}}(\\Lambda_1)$ the set of $\\Lambda_2 \\in \\mathcal{V}_{\\alpha_{M}}^{ \\mathrm{reg}}(\\Lambda_1)$ such that $(\\alpha, \\Lambda \\to \\Lambda_2)$ is regular.\n\nLet $W_\\alpha^+:= M_{\\max \\mathsf{f}_\\alpha} \\setminus M_{\\mathsf{f}_\\alpha}$ and $W_\\alpha^-:= M_{ \\mathsf{f}_\\alpha} \\setminus M$.\nSince the Lagrangians $L_1$ and $L_2$ are conical in $M_{\\max \\mathsf{f}_\\alpha} \\setminus M$ we obtain for elements $\\Lambda_2 \\in \\mathcal{V}_{\\alpha_{(\\Sigma,\\lambda)}}^{ \\alpha-\\mathrm{reg}}(\\Lambda_1)$ transfer maps\n\\\\\n\\[\n\\begin{CD}\n \\mathrm{HW}(M_{\\max \\mathsf{f}_\\alpha}, L\\to L_2) @>\\Phi_{W^+,L \\to L_2}>> \\mathrm{HW}(M_{\\mathsf{f}_\\alpha},L \\to L_2) @>\\Phi_{W^-,L \\to L_2}>> \\mathrm{HW}(M,L \\to L_2).\n\\end{CD}\n\\]\n\\\\\nBy Lemma \\ref{lem:changeofhypersurface}, the composition $\\Phi_{W^-,L \\to L_2}\\circ \\Phi_{W^+,L \\to L_2}$ is induced by asymptotic morphisms, and the f.d.s. $ \\widetilde{\\mathrm{HW}}(M_{\\mathsf{f}_\\alpha},L \\to L_2)$ and $\\widetilde{\\mathrm{HW}}(M,L \\to L_2)$ are $(\\max \\mathsf{f}_{\\alpha}, 1)$- interleaved.\n\nThe following proposition then follows from combining this observation and Proposition \\ref{uniformmodules2}.\n\\begin{prop} \\label{uniformgrowth}\nLet $\\alpha$ be a contact form on $(\\Sigma,\\xi_{M})$ and assume that the function $\\mathsf{f}_\\alpha$ defined by $\\alpha=\\mathsf{f}_\\alpha \\alpha_{M}$ is $\\geq 1$.\nThen, the family of f.d.s. $(\\widetilde{\\mathrm{HW}}(M_{\\mathsf{f}_\\alpha},L \\to L_2))_{\\Lambda_2 \\in \\mathcal{V}_{\\alpha_{M}}^{\\alpha-\\mathrm{reg}}(\\Lambda_1)}$ satisfies\n\\begin{equation} \\label{equniformgrowth1}\n\\Gamma^{\\mathrm{symp}}_{\\Lambda_2 \\in \\mathcal{V}_{\\alpha_{M}}^{\\alpha-\\mathrm{reg}}(\\Lambda_1)}(M_{\\mathsf{f}_\\alpha},L \\to L_2) \\geq \\frac{\\Gamma^{\\mathrm{symp}}(M, L)}{\\max \\mathsf{f}_\\alpha}.\n\\end{equation} \n\\end{prop}\n\n\nA reasoning identical to the one used to establish \\eqref{equsefulestimate} shows that for every $\\Lambda_2 \\in \\mathcal{V}_{\\alpha_{M}}^{\\alpha-\\mathrm{reg}}(\\Lambda_1)$ and for the exact filling $L'_2$ of $\\Lambda_2$ constructed above we have \n\\begin{equation}\n\\mathrm{HW}^a(M_{\\mathsf{f}_\\alpha}, L \\to L_2) \\mbox{ and } \\mathrm{HW}^a(M_{\\mathsf{f}_\\alpha}, L \\to L'_2) \\mbox{ are isomorphic.}\n\\end{equation}\nCombining this with Proposition \\ref{uniformgrowth} we have\n\\begin{cor} \\label{coro:uniformgrowth}\nLet $\\alpha$ be a contact form on $(\\Sigma,\\xi_{M})$ and assume that the function $\\mathsf{f}_\\alpha$ defined by $\\alpha=\\mathsf{f}_\\alpha \\alpha_{M}$ is $\\geq 1$.\nThen, the family of f.d.s. $(\\widetilde{\\mathrm{HW}}(M_{\\mathsf{f}_\\alpha},L \\to L'_2))_{\\Lambda_2 \\in \\mathcal{V}_{\\alpha_{M}}^{\\alpha-\\mathrm{reg}}(\\Lambda_1)}$ satisfies\n\\begin{equation} \\label{equniformgrowth1}\n\\Gamma^{\\mathrm{symp}}_{\\Lambda_2 \\in \\mathcal{V}_{\\alpha_{M}}^{\\alpha-\\mathrm{reg}}(\\Lambda_1)}(M_{\\mathsf{f}_\\alpha},L \\to L'_2) \\geq \\frac{\\Gamma^{\\mathrm{symp}}(M, L)}{\\max \\mathsf{f}_\\alpha}.\n\\end{equation} \n\\end{cor}\n\nRecall that for every $\\Lambda_2 \\in \\mathcal{V}_{\\alpha_{M}}^{\\alpha-\\mathrm{reg}}(\\Lambda_1)$ the exact filling $L'_2$ of $\\Lambda_2$ satisfies\n\\begin{equation} \\label{bound}\n \\#(L'_2 \\cap L) = \\mathsf{C}_{\\mathrm{regium}}.\n\\end{equation}\n\nNow, given a Legendrian $\\Lambda_2 \\in \\mathcal{V}_{\\alpha_{M}}^{\\alpha-\\mathrm{reg}}(\\Lambda_1)$ let $\\mathsf{N}_\\alpha^a(\\Lambda \\to \\Lambda_2)= \\# \\mathcal{T}^a_{\\Lambda \\to \\Lambda_2}(\\alpha)$.\nWe define \n\\begin{equation}\n\\mathsf{N}^a_\\alpha( \\Lambda \\to \\mathcal{V}_{\\alpha_{M}}^{\\alpha-\\mathrm{reg}}(\\Lambda_1)):= \n\\inf_{\\Lambda_2 \\in \\mathcal{V}_{\\alpha_{M}}^{\\alpha-\\mathrm{reg}}(\\Lambda_1)} \\{\\mathsf{N}^a_\\alpha(\\Lambda \\to \\Lambda_2) \\}.\n\\end{equation}\n\nLet $a > \\epsilon$. By the results of Section \\ref{rem:rem1} there exists a Hamiltonian $H^a\\in \\mathcal{H}_{\\mathrm{reg}}(M_{\\mathsf{f}_\\alpha},L \\to L'_2)$ with slope $0$ around the origin. We endow $\\mathbb{D}^n(\\rho)$ with the Euclidean metric, and consider on $T^*_1\\mathbb{D}^n(\\rho) = \\mathbb{D}^n(\\rho) \\times S^{n-1}$ the contact form $\\alpha_{\\euc}$ associated to the Euclidean metric. For each $z\\in \\mathbb{D}^n(\\rho)$ the sphere $S^{n-1}_z:= \\{z\\} \\times S^{n-1}$ is Legendrian in $(\\mathbb{D}^n(\\rho) \\times S^{n-1},\\ker \\alpha_{\\euc}) $. \nLet $g_{\\round}$ be the metric with constant curvature $1$ on $S^{n-1}$ and $g_{\\euc}$ be the Euclidean metric on $\\mathbb{D}^{n}(\\rho)$. The metric $\\widetilde{g}= g_{\\euc} \\oplus g_{\\round}$ on $\\mathbb{D}^n(\\rho) \\times S^{n-1}$ is compatible with the contact form $\\alpha_{\\euc}$; see \\cite{calvaruso}.\n \n\n\n\\begin{prop} \\label{propgrowth}\nLet $\\alpha$ be a contact form on $(\\Sigma,\\xi_{M})$ and assume that we have $\\Gamma^{\\mathrm{alg}}_S(M_{f_\\alpha}, L) >0$. Then there exists a Riemannian metric $g$ on $(\\Sigma,\\xi_{M})$ adapted to the $\\alpha$, such that\n\\begin{equation}\n\\limsup_{t \\to +\\infty} \\frac{ \\log \\Vol_g^{n-1}(\\phi_{\\alpha}^t(\\Lambda))}{t} \\geq \\frac{\\Gamma^{\\mathrm{symp}}(M, L)}{\\max f_\\alpha}>0,\n\\end{equation}\nwhere $\\Vol_g^{n-1}$ is the $(n-1)$-dimensional volume with respect to $g$, and $f_\\alpha$ is the function such that $\\alpha=f_\\alpha \\alpha{(\\Sigma,\\lambda)}$.\n\\end{prop}\n\\textit{Proof:} \nThe proof will consist of several steps. \n\n\\textbf{Step 1.} It suffices to prove the proposition for all contact forms $\\alpha$ for which $\\mathsf{f}_\\alpha\\geq1$. \nIndeed assume that the proposition holds for all such contact forms.\n\nTake a contact form ${\\alpha'}$ on $(\\Sigma,\\xi_{M})$. For the contact form $\\widehat{\\alpha}:= \\frac{\\alpha'}{\\min \\mathsf{f}_{\\alpha'}}$ we have $\\mathsf{f}_{\\widehat{\\alpha}}= \\frac{\\mathsf{f}_{\\alpha'}}{\\min \\mathsf{f}_{\\alpha'} }\\geq 1$. By assumption there is a Riemannian metric $g$ on $\\Sigma$ compatible with $\\widehat{\\alpha}$ and such that \n\\begin{equation}\n\\limsup_{t \\to +\\infty} \\frac{ \\log \\Vol_g^{n-1}(\\phi_{\\widehat{\\alpha}}^t(\\Lambda))}{t} \\geq \\frac{\\Gamma^{\\mathrm{symp}}(M, L)}{\\max \\mathsf{f}_{\\widehat{\\alpha}}}.\n\\end{equation}\n\nThe Riemannian metric $g':=(\\min \\mathsf{f}_{{\\alpha'}})^2g$ is compatible with $\\alpha'$. \nA simple computation shows that \n$\\limsup_{t \\to +\\infty} \\frac{ \\log \\Vol_{g'}^{n-1}(\\phi_{{\\alpha'}}^t(\\Lambda))}{t} \\geq \\frac{\\Gamma^{\\mathrm{symp}}(M, L)}{\\max \\mathsf{f}_{{\\alpha'}}}$, as claimed.\nWe thus fix from now on a contact form $\\alpha$ on $(\\Sigma,\\xi_{M})$ with $\\mathsf{f}_\\alpha\\geq1$.\n\n\\textbf{Step 2. A tubular neighbourhood of $\\Lambda_1$ and construction of the metric $g$} \\\\\nIt follows from the Legendrian neighbourhood theorem (see \\cite[Proposition 43.18]{MichorKriegl}) that there exists a tubular neighbourhood $\\mathcal{V}(\\Lambda_1)$ of $(\\Lambda_1)$ and a contactomorphism \n$\\Upsilon : (\\mathcal{V}(\\Lambda_1), \\xi_{M}) \\to (\\mathbb{D}^n(\\rho) \\times S^{n-1},\\ker \\alpha_{\\euc}) $ that satisfies\n\\begin{eqnarray} \\label{neighbourhood}\n\\Upsilon^* \\alpha_{\\euc} = \\alpha, \\\\\n\\Upsilon(\\Lambda_1) = \\{0\\} \\times S^{n-1}.\n\\end{eqnarray}\n\nWe extend the Riemannian metric $\\Upsilon^*\\widetilde{g}$, which is compatible with $\\alpha$ on $\\mathcal{V}(\\Lambda_1)$, to a metric $g$ on $\\Sigma$ which is compatible with the contact form $\\alpha$.\n\n\n\nAfter shrinking the neighbourhood $\\mathcal{V}(\\Lambda_1)$ and $\\rho>0$, we can assume that for every $z \\in \\mathbb{D}^{n}(\\rho)$ the Legendrian $\\Lambda^z := \\Upsilon^{-1}(\\{z\\} \\times S^{n-1})$ is in the neighbourhood $\\mathcal{V}_{\\alpha_{M}}(\\Lambda_1)$ constructed in Section~\\ref{subsec:technical}. \n\n\n\n\\textbf{Step 3.} \nFor each $a>0$ we define the map $F^a_{\\Lambda}: \\Lambda \\times [0,a] \\to \\Sigma$ by\n\\begin{equation}\nF^a_{\\Lambda}(q,t) = \\phi^t_\\alpha(q).\n\\end{equation}\nLet $\\Cyl^a_\\alpha(\\Lambda)$ be the image $F^a_{\\Lambda}(\\Lambda \\times [0,a] )$.\nWe want to estimate from below the $n$-dimensional volume $\\Vol^{n}_g(\\Cyl^a_\\alpha(\\Lambda))$ of $\\Cyl^a_\\alpha(\\Lambda)$ with respect to the Riemannian metric $g$. For this we define $\\mathfrak{B}^a_\\alpha(\\Lambda):= \\Upsilon(\\Cyl^a_\\alpha(\\Lambda) \\cap \\mathcal{V}(\\Lambda_1)) $. We have\n\\begin{equation} \\label{tititi}\n\\Vol^{n}_g(\\Cyl^a_\\alpha(\\Lambda)) \\geq \\Vol^{n}_{{g}}(\\Cyl^a_\\alpha(\\Lambda)\\cap \\mathcal{V}(\\Lambda_1)) = \\Vol^{n}_{\\widetilde{g}}(\\mathfrak{B}^a_\\alpha(\\Lambda)).\n\\end{equation}\n\n\nLet $\\Pi: \\mathbb{D}^{n}(\\rho) \\times S^{n-1} \\to \\mathbb{D}^{n}(\\rho)$ be the projection to the first coordinate. \nApplying Sard's Theorem to the map $\\Pi \\circ \\Upsilon \\circ F^a_\\Lambda: (\\{a\\} \\times \\Lambda) \\cap (F^a_\\Lambda)^{-1}( \\mathcal{V}(\\Lambda_1)) \\to \\mathbb{D}^n(\\rho)$ we conclude that the set $ \\mathbb{D}^{n}(\\rho) \\setminus \\Pi\\circ \\Upsilon(\\phi^a_\\alpha(\\Lambda))$ is an open set of full Lebesgue measure in $ \\mathbb{D}^{n}(\\rho)$.\nWe define the set $\\mathfrak{U}^a_\\alpha(\\Lambda) \\subset \\mathbb{D}^{n}(\\rho) \\setminus \\Pi\\circ \\Upsilon(\\phi^a_\\alpha(\\Lambda) )$ by the property\n\\begin{itemize}\n\\item $z \\in \\mathfrak{U}^a_\\alpha(\\Lambda)$ if all $\\alpha$-Reeb chords from $\\Lambda$ to $\\Lambda^z$ with length $0$, we can fix $0<\\eta <\\frac{\\Gamma^{\\mathrm{symp}}(M, L)}{\\max \\mathsf{f}_\\alpha}$. It follows from Corollary \\ref{corocrucial} that there exists a sequence $a_j \\to +\\infty$ such that $h^{a_j}(z) \\geq e^{\\eta a_j}$ for all $z \\in \\widetilde{\\mathfrak{U}}^{a_j}_\\alpha(\\Lambda)$. Since $\\widetilde{\\mathfrak{U}}^{a_j}_\\alpha(\\Lambda)$ is dense in ${\\mathfrak{U}}^{a_j}_\\alpha(\\Lambda)$ and $h^{a_j}$ is locally constant on ${\\mathfrak{U}}^{a_j}_\\alpha(\\Lambda)$ we obtain $h^{a_j}(z) \\geq e^{\\eta a_j} \\mbox{ for all } z \\in {\\mathfrak{U}}^{a_j}_\\alpha(\\Lambda)$\nand all $a_j$.\nWith \\eqref{tititi} it follows that \n\\begin{equation} \\label{volgrowth}\n\\Vol^{n}_g(\\Cyl^{a_j}_\\alpha(\\Lambda)) \\geq \\int_{ \\mathfrak{U}^{a_j}_\\alpha(\\Lambda)} h^{a_j}(z) \\ d \\mathsf{vol}_{g_{\\euc}} \\geq e^{\\eta a_j} \\uppi \\rho^2\n\\end{equation}\nfor every $a_j$.\n\n\\textbf{Step 5. A Fubini type equality}\nWe define $\\widehat{g}:= (F^a_\\Lambda)^*g$. Then \n\\begin{equation}\\label{changecoord}\n\\Vol^{n}_g(\\Cyl^{a}_\\alpha(\\Lambda))= \\int_{ \\Lambda \\times [0,a]} d \\mathsf{vol}_{\\widehat{g}},\n\\end{equation}\nwhere $d \\mathsf{vol}_{\\widehat{g}}$ is the volume form associated to $\\widehat{g}$.\nSince the metric $g$ is adapted to the contact form $\\alpha$ the Reeb vector field has length $1$ and is orthogonal to the Legendrian spheres $F^a_\\Lambda(t,\\Lambda) = \\phi^t_\\alpha(\\Lambda)$ for every $t\\in [0,a]$.\nLetting $\\partial_t$ be the tangent vector field on $[0,a] \\times \\Lambda$ associated to the first coordinate $t\\in [0,a]$, and using the definition of $F^a_\\Lambda$, it follows that $D(F^a_\\Lambda) \\partial_t = X_\\alpha$. Therefore $\\partial_t$ has $\\widehat{g}$-norm equal to $1$ at every point in $[0,a] \\times \\Lambda$, and is orthogonal to the spheres $\\{t\\}\\times \\Lambda$. We thus conclude that \n\\begin{equation} \\label{fubini}\n \\Vol^{n}_g(\\Cyl^{a}_\\alpha(\\Lambda))=\\int_{ \\Lambda \\times [0,a]} d \\mathsf{vol}_{\\widehat{g}} = \\int_0^{a} \\Vol_{\\widehat{g}}^{n-1}(\\{t\\} \\times \\Lambda) dt = \\int_0^a \\Vol_{g}^{n-1}( \\phi^t_\\alpha(\\Lambda)) dt,\n\\end{equation}\nwhere $\\Vol_{\\widehat{g}}^{n-1}$ is the $(n-1)$-dimensional volume associated to $\\widehat{g}$.\n\n\n\\textbf{Step 6. End of the proof.}\nTo finish the proof we argue by contradiction and assume that \n$\\limsup_{t \\to +\\infty } \\frac{\\log \\Vol_{g}^{n-1}( \\phi^t_\\alpha(\\Lambda))}{t} < \\eta$. In this case, there exist $ a_0 >0$ and $\\varepsilon>0$ such that for all $t\\geq a_0$ we have $\\Vol_{g}^{n-1}( \\phi^t_\\alpha(\\Lambda)) \\leq e^{t(\\eta-\\varepsilon)}$. \nIntegrating both sides of this inequality from $0$ to $a\\geq a_0$ and invoking \\eqref{fubini} we obtain\n\\begin{equation} \\label{cacaca}\n\\Vol^{n}_g(\\Cyl^{a}_\\alpha(\\Lambda))\\leq \\frac{e^{a (\\eta-\\varepsilon)} - e^{a_0 (\\eta-\\varepsilon)}}{\\eta-\\varepsilon} + \\int_0^{a_0} \\Vol_{g}^{n-1}( \\phi^t_\\alpha(\\Lambda)) dt.\n\\end{equation}\nFor $a$ large enough the right hand side of \\eqref{cacaca} is smaller than $e^{\\eta a}\\uppi \\rho^2$, contradicting \\eqref{volgrowth}. We thus conclude that \n\\begin{equation}\n\\limsup_{t \\to +\\infty } \\frac{\\log \\Vol_{g}^{n-1}( \\phi^t_\\alpha(\\Lambda))}{t} \\geq \\eta.\n\\end{equation}\n\nSince this is valid for any $\\eta < \\frac{\\Gamma^{\\mathrm{symp}}(M, L)}{\\max \\mathsf{f}_{\\alpha}}$, the proof of the proposition is completed. \\qed\n\n\\textit{\\textbf{Proof of Theorem \\ref{theorementropy}:}} \\\\\nFrom Proposition \\ref{propgrowth} and Yomdin's theorem (see \\eqref{Yomdin}) it follows that if $\\Gamma^{\\mathrm{symp}}(M,L)>0$, then for every contact form $\\alpha$ on $(\\Sigma,\\xi_{M})$ we have\n\\begin{equation} \\label{entropyineq}\nh_{\\mathrm{top}}(\\phi_{\\alpha}) \\geq \\frac{\\Gamma^{\\mathrm{symp}}(M,L)}{\\max(\\mathsf{f}_{\\alpha})}.\n\\end{equation}\nWe then obtain Theorem \\ref{theorementropy} by combining \\eqref{entropyineq} with the inequality\n\\begin{equation*}\n\\Gamma^{\\mathrm{symp}}(M,L) \\geq \\frac{\\Gamma^{\\mathrm{alg}}_S(M,L)}{\\rho(S)}.\n\\end{equation*} \\color{black}\n from Lemma \\ref{prop:alg_symp}. \\color{black}\n\\qed\n\n\\color{black}\n\n\\section{Algebras in loop space homology}\\label{sec:ring_module} \n\n\\color{black}\n\n\nLet $V$ be a compact manifold and fix a point $q \\in V$.\nWe denote by $\\Omega_q(V)$ the based loop space of $V$ with basepoint in $q$, which is the space of continuous maps from $[0,1]$ to $V$ that map $0$ and $1$ to $q$. \n\nThe concatenation of based loops gives $\\Omega_q(V)$ the structure of an $H$-space (see \\cite{Hatcher}). \nMore precisely, the concatenation induces the so-called Pontrjagin product on the singular homology $\\mathrm{H}_*(\\Omega_{q}(V))$ of $\\Omega_{q}(V)$ with $\\mathbb{Z}_2$ coefficients. \\color{black}\nThe Pontrjagin product $[a_1]\\cdot[a_2]$ of two homology classes $[a_1],[a_2] \\in \\mathrm{H}_*(\\Omega_{q}(V)$ is well-known to be associative.\nAs it is distributive with respect to the vector space structure of $\\mathrm{H}_*(\\Omega_{q}(V))$, it makes $\\mathrm{H}_*(\\Omega_{q}(V))$ into a ring. Because the homology $\\mathrm{H}_*(\\Omega_{q}(V))$ is considered with coefficients in $\\Z_2$ it actually has the structure of an algebra. \n\n\n \n \n\\subsection{Relation between the algebra structure of the singular homology of loop spaces and the algebra structures of the Floer homology of cotangent bundles} \\color{black}\n Given a manifold $V$ and $q \\in V$ we denote by $L_q \\subset T^* V$ the cotangent fibre over $q$.\nThe singular homology $\\mathrm{H}_*(\\Omega_{q}(V))$ of the based loop space $\\Omega_{q}(V)$ is isomorphic to the wrapped Floer homology $\\mathrm{HW}(T^*V,L_q)$; see Viterbo \\cite{Viterbo1999}, Salamon-Weber \\cite{SW} and Abbondandolo-Schwarz \\cite{AS-iso} for different proofs. \n\nThe Floer homology $\\mathrm{HW}(H_g,L_q)$ is isomorphic to the wrapped Floer homology $\\mathrm{HW}(L_q)$ we use in this paper. The key point is that the Hamiltonian $H_g$ is quadratic in the fibres. This isomorphism is proven in \\cite{Ritter2013}, and it preserves the triangle product and the spectral value of homology classes.\n\nLet $\\Psi_{AS,q}: \\mathrm{H}_*(\\Omega_{q}(V)) \\to \\mathrm{HW}(T^*V,L_q)$ be the isomorphism constructed in \\cite{AS-iso}.\nIn \\cite{AS-product} the authors proceed to study more properties of the map $\\Psi_{AS,q}$. \\color{black}\n They show that $\\Psi_{AS,q}$ is also algebra isomorphism if we consider $\\mathrm{H}_*(\\Omega_{q}(V))$ as an algebra with the Pontrjagin product and $\\mathrm{HW}(H_g,L_q)$ as an algebra with the triangle product. Combining this with the isomorphism $\\mathrm{HW}(H_g,L_q) \\cong\\mathrm{HW}(L_q)$ we obtain the following \n\\begin{thm}[Abbondandolo-Schwarz \\cite{AS-product}] \\label{ASring}\nThe singular homology $\\mathrm{H}_*(\\Omega_{q}(V))$ and the wrapped Floer homology $\\mathrm{HW}(L_q)$ are isomorphic as algebras. \n\\end{thm} \\color{black}\nFor simplicity we will still denote by $\\Psi_{AS,q}$ the isomorphism between $\\mathrm{H}_*(\\Omega_{q}(V))$ and $\\mathrm{HW}(L_q)$.\n\n\\section{Topological operations}\\label{sec:top_operations}\n\n\n\n\\subsection{Subcritical surgery}\\label{subsec:subcritical}\n\nHere we study the Viterbo transfer maps under subcritical handle attachment in the situation that is sufficient for our purpose, that is we assume that the Lagrangians do not intersect the handle. \n\n\\color{black} \nLet $W=(Y_W,\\omega,\\lambda)$ be a Liouville domain, $\\Sigma = \\partial W$, $\\lambda|_{\\Sigma} = \\alpha$ and $\\xi = \\ker \\alpha$. We recall some notions using the terminology of \\cite[Section 2.5.2]{GeigesBook}. The form $d\\alpha$ endows $\\xi$ with a natural conformal symplectic bundle structure.\nLet $S$ be an isotropic submanifold of $(\\Sigma,\\xi)$. We write $TS^{\\bot}$ for the sub-bundle of $\\xi$ that is $d\\alpha$-orthogonal to $TS$. Because $S$ is isotropic $TS \\subset TS^{\\bot}$.\nWe can therefore write the normal bundle of $S$ in $\\Sigma$ as \n\\[\nT \\Sigma \/T S = T\\Sigma\/\\xi \\oplus \\xi\/TS^{\\bot} \\oplus TS^{\\bot} \/ TS.\n\\] \\color{black}\nThe \\textit{conformal symplectic normal bundle} $\\mathrm{CSN(S)} = TS^{\\bot} \/ TS$ has a natural conformal symplectic structure via $d\\alpha$. If $S$ is a sphere, $T\\Sigma\/\\xi \\oplus \\xi\/TS^{\\bot}$ has a trivialization. The following theorem is due to Weinstein.\n\\begin{thm}\\cite{Weinstein1990}\nLet $S^n$ be an isotropic sphere in $\\Sigma$ with a trivialization of $\\mathrm{CSN}(S)$. Then there is a Liouville domain $M$ with an exact embedding $W \\subset M$, such that $\\partial M$ is obtained from $\\Sigma$ by surgery on $S$. \n\\end{thm}\n\nThe Liouville domain $M$ is obtained by attaching an $(n+1)$-handle to $W$ and the Liouville vector field $X$ can by chosen such that there is exactly one point $p \\in M\\setminus W$ where $X$ vanishes. The integral lines of $X$ that are asymptotic to $p$ intersect $\\Sigma$ in $S$ and $\\partial M$ in the co-core sphere $B \\subset \\partial M$. (See \\cite{Weinstein1990, Cieliebak2001} or \\cite[Chapter 6]{GeigesBook} for details.) \n\nLet now $L^{'}_0$, $L^{'}_1$ be two asymptotically conical exact Lagrangians in $ W$ whose boundaries $\\Lambda^{'}_0$ and $\\Lambda^{'}_1$ in $\\Sigma$ do not intersect $S$.\nOutside $S$ the integral lines of the Liouville vector field starting at $\\partial W$ intersect $\\partial M$ and so the completed Lagrangians $\\widehat{L^{'}_{i}} \\subset \\widehat{M}$ intersect $\\partial M$. Moreover, $L_{i} = \\widehat{L^{'}_{i}} \\cap M \\subset M$ for $i=0,1$ are exact and conical in the complement of $W$. We say that $(M,L_0,L_1)$ is \\textit{obtained by surgery} from $(W,L^{'}_0,L^{'}_1).$ \n\n \n\nAs described in section \\ref{sec:Viterbo} we get a Viterbo transfer map\n\\[\n{j}_{!}(L_0,L_1): \\widetilde{\\mathrm{HW}}(M,L_0 \\to L_1) \\rightarrow \\widetilde{\\mathrm{HW}}(W,L^{'}_0 \\to L^{'}_1). \n\\]\n\n\nAssume that the isotropic sphere $S$ has the property that there is no Reeb chord from $\\Lambda^{'}_0$ to $S$.\nIf $S$ is subcritical, i.e. $\\dim(S) < n-1$, this can be achieved by a generic choice of $S$. \n\nThe following proposition was proved by Cieliebak (\\cite{Cieliebak2001}) for symplectic homology. The proof in our situation is analogous and even simpler. We give it here for the convenience of the reader. \n\\begin{prop}\\label{Viterbo_iso}\nThe Viterbo transfer map in the direct limit,\n\\[\n\\bar{j}_{!}(L_0,L_1): \\mathrm{HW}(M,L_0,L_1) \\to \\mathrm{HW}(W,L^{'}_0, L^{'}_1), \n\\]\nis an isomorphism.\n\\end{prop} \n\n\\input{figure2.tex}\n\n\nFor the proof of Proposition \\ref{Viterbo_iso} it is convenient to introduce the following weaker form of interleaving of f.d.s. \nLet $\\sigma: [0,\\infty) \\rightarrow [0,\\infty)$ be a monotone increasing function and $V$ a filtered directed system. \\color{black} Analogously to the notation in \\ref{subsubsec:fds} let $(V(\\sigma), \\pi(\\sigma))$ be given by $V({\\sigma})_t = V_{\\sigma (t) t}$, $\\pi(\\sigma)_{s\\rightarrow t}= \\pi_{\\sigma(s) s \\rightarrow \\sigma(t) t}$ and $\\pi[\\sigma]_t = \\pi_{\\sigma(t) t}$. If $f$ is a morphism from $(V,\\pi)$ to another f.d.s. we write $f(\\sigma)_t = f_{\\sigma(t) t}$ for the induced morphism with domain $(V(\\sigma), \\pi(\\sigma))$. \\color{black} \nCall two f.d.s. $(V,\\pi_V)$ and $(W,\\pi_W)$ \\textit{weakly interleaved} if there are morphisms\n$f:V\\rightarrow W(\\sigma_1)$ and $g:W \\rightarrow V(\\sigma_2)$ for monotone increasing functions $\\sigma_1, \\sigma_2 \\geq 1$ \nsuch that \n\\[f(\\sigma_2)\\circ g = \\pi_{W}[\\widetilde{\\sigma}_1] \\text{ and } g(\\sigma_1)\\circ f = \\pi_{V}[\\widetilde{\\sigma}_2],\\]\nwhere $\\widetilde{\\sigma}_1$, and $\\widetilde{\\sigma}_2$ are suitably chosen.\nThe fact that the map $\\bar{j}_{!}(L_0,L_1)$ in Proposition \\ref{Viterbo_iso} is an isomorphism will follow from a weak interleaving of the corresponding f.d.s., which is in general not an interleaving. This is the reason why we cannot directly prove lower bounds for $\\Gamma^{\\mathrm{symp}}(M,L_0 \\to L_1)$ in terms of $\\Gamma^{\\mathrm{symp}}(W,L^{'}_0 \\to L^{'}_1)$ and this was originally our motivation to introduce the algebraic growth of wrapped Floer homology. \n\n\n\\textit{Proof of Proposition \\ref{Viterbo_iso}: }\nLet $U = \\widetilde{\\mathrm{HW}}(M,L_0 \\to L_1)$, and $V =\\widetilde{\\mathrm{HW}}(W,L^{'}_0 \\to L^{'}_1)$. \nWe will construct a filtered directed system $Q$ that is isomorphic to $V$ and weakly interleaved with $U$. \n\nFor convenience we may assume $\\mathsf{K}(M,L_0 \\to L_1) = 0$. \nLet $S \\subset \\partial W$ be the attaching sphere and $B \\subset \\partial M$ be the co-core sphere. \nFor $a > 0$, choose a tubular neighbourhood $U_a \\subset \\partial W$ of $S$ such that there is no Reeb trajectory starting at $\\Lambda^{'}_0$ that intersects $U_a$ at a time less than $a$, and such that $U_{b} \\subset U_a$ if $aa$ and $\\partial W \\setminus U_a \\subset \\partial M_{f_a}$.\nDefine $\\sigma(a) = \\frac{1}{\\min_{\\partial M} f_a}$.\nDefine $Q_a = \\mathrm{HW}^a(M_{f_a}, L_0 \\to L_1)$, where by abuse of notation we write $L_i$ instead of $L_i \\cap M_{f_a}$, $i= 0,1$. \nFor $a < b$ define $\\pi_{a\\to b} : Q_a \\to Q_b$ as the composition of the Viterbo map $\\mathrm{HW}^a(M_{f_a}, L_0 \\to L_1) \\to \\mathrm{HW}^a(M_{f_b}, L_0 \\to L_1)$ and the persistence map $\\mathrm{HW}^a(M_{f_b}, L_0 \\to L_1) \\to \\mathrm{HW}^b(M_{f_b}, L_0 \\to L_1)$.\nBy the commutativity of the Viterbo map with persistence maps and by functoriality of the Viterbo map\n it follows that $\\pi_{a\\to c} = \\pi_{b\\to c} \\circ \\pi_{a \\to b}$, for $a 0$. Assume that $L_0$ and $L_1$ are conical in the complement of $W_{\\frac{1}{2}}$. \nLet $H_{\\mu}$ be an admissible Hamiltonian with slope $\\mu$ with respect to $W_{\\frac{1}{2}}$. Consider a Hamiltonian $K_{\\mu}$ such that\n\\begin{align}\nK_{\\mu}(x) = &H_{\\mu}(x), \\text{ if } x \\in W_{\\frac{3}{4}}. \\\\\nK_{\\mu}(x) = &H_{\\mu}(x), \\text{ if } x = (r,y) \\in (0, +\\infty) \\times \\partial W\\setminus U_a. \\label{K2}\n\\end{align}\nIt follows that $K_{\\mu}(x) = 2\\mu r + b$, for some $b\\in \\R$, where $x$ is written in the coordinates $(r,y) \\in (1, \\infty) \\times \\partial W \\setminus U_a$. \nHence we can assume additionally that \n\\begin{align}\nK_{\\mu}(x) = &2\\mu r + b, \\text{ where } x = (r,y) \\in (1, \\infty) \\times \\partial M_{f_a}. \\label{K3} \n\\end{align}\nBy definition of $U_a$, $\\mathcal{A}_{H_{\\mu}}$ and $\\mathcal{A}_{K_{\\mu}}$ have the same critical points, and so it follows from \\cite[Lemma 7.2]{AbouzaidSeidel2010} that we actually have $\\mathrm{HW}(K_{\\mu}) = \\mathrm{HW}(H_{\\mu})$. On the other hand $K_{\\mu} - {\\frac{1}{2}}\\mu$ is admissible with respect to $M_{f_a}$ with slope $2\\mu$. \\color{black} \n One concludes, reasoning as in Lemma \\ref{M_epsilon}, that \n$Q_a = \\mathrm{HW}^{a}(M_{f_a}, L_0 \\to L_1) \\cong \\mathrm{HW}^{\\frac{1}{2}a}(W_{\\frac{1}{2}},L^{'}_0 \\to L^{'}_1)$ which is, by Lemma \\ref{M_epsilon}, isomorphic to $\\mathrm{HW}^{a}(W,L^{'}_0 \\to L^{'}_1) = V_a$.\nThat this identification respects the persistence morphisms of $Q$ and $V$ is again deduced from the functoriality of the Viterbo maps and the fact that the Viterbo maps are themselves morphisms of filtered directed systems.\nDenote the isomorphism from $Q$ to $V$ by $\\tau$. We have obtained a weak interleaving $(\\tau \\circ \\phi, \\psi \\circ \\tau^{-1})$. Moreover $\\tau \\circ \\phi = {j}_{!}$ by construction. \\color{black} \n\\qed\n\n\n\\subsection{Plumbing}\n\nLet $Q_1$ and $Q_2$ be closed orientable $n$-dimensional manifolds. We let $D^*Q_i$ be the unit cotangent bundle of $Q_i$. We choose balls $B_i \\subset Q_i$ in each $Q_i$.\nThe plumbing $N$ of $D^*Q_1$ and $D^*Q_2$ is obtained by identifying $D^*B_1$ and $D^*B_2$ via a symplectomorphism that swaps the momentum and position coordinates of these manifolds; see \\cite{AbouzaidSmith2012,GeigesBook} for the details. \nThere are obvious embeddings of $D^*(Q_1 \\setminus B_1)$ and $D^*(Q_2 \\setminus B_2)$ into $N$.\n It is shown in \\cite[Section 4]{AbouzaidSmith2012} that $N$ admits a Liouville structure which coincides with those of $D^*(Q_i \\setminus B_i)$ on the image of these embeddings. This implies that for points $q_1 \\in Q_1\\setminus B_1$ the cotangent disc fibre $L_{q_1}$ over $q_1$ survives as a conical exact Lagrangian in the Liouville domain $N$.\n\nThis construction can be generalised in the following way. Let $\\{Q_i \\mid 1 \\leq i \\leq k\\}$ be a finite collection of orientable $n$-dimensional manifolds. Let $\\mathsf{T}$ be a tree with $k$ vertices and use a bijection to associate to each vertex a manifold $Q_i$. For each edge $\\eta$ leaving the ``vertex'' $Q_i$ we choose an embedded open ball $B_i(\\eta)$ in $Q_i$. We assume that these balls are chosen to be disjoint and do not cover $Q_i$. For all $i\\neq j$ and every edge $\\eta$ connecting $Q_i$ and $Q_j$ (there can be at most one such edge as $\\mathsf{T}$ is a tree) we identify $D^*(B_i(\\eta))$ and $D^*(B_j(\\eta))$ by the recipe explained in the previous paragraph. The resulting manifold $N$ can be given a Liouville structure as explained in \\cite[Section 4]{AbouzaidSmith2012} and \\cite{GeigesBook}. Let $\\dot{Q}_1$ be the complement of the ``edge balls'' in $Q_1$, and $q_1 \\in \\dot{Q}_1$.\n In \\cite[Section 4]{AbouzaidSmith2012} the following result is proved.\n \\begin{thm} \\cite{AbouzaidSmith2012} \\label{thm:plumbing}\n There exists an injective algebra homomorphism from the group algebra $\\mathbb{Z}_2[\\pi_1(Q_1)]$ to $\\mathrm{HW}(N,L_{q_1})$.\n \\end{thm}\nIn fact the injective algebra homomorphism obtained in \\cite{AbouzaidSmith2012} is for the respective homologies with $\\mathbb{Z}$ coefficients, and applying the Universal Coefficient Theorem one obtains the homomorphism mentioned above. Thus if $\\pi_1(Q_1)$ grows exponentially then $\\mathrm{HW}(N,L_{q_1})$ has exponential algebraic growth; see Section \\ref{preliminaries}.\n\n\\textit{Proof of Proposition \\ref{prop:operations}}: \nPart A) follows from Proposition \\ref{Viterbo_iso} and Part B) follows from Theorem \\ref{thm:plumbing}. \\qed\n\n\\section{Construction of contact structures with positive entropy }\\label{sec:constructions}\n\nIn this section we prove Theorem \\ref{spheres} and Theorem \\ref{maincorollary}. \n\n\\subsection{Preliminaries} \\label{preliminaries}\n\n\nLet $Q$ be a closed connected smooth manifold and $g$ a Riemannian metric on $Q$. Let $(D_{g}^*Q,\\lambda_{geo}) \\subset (T^*Q,\\lambda_{geo})$ be the unit disk bundle with respect to the Riemannian metric $g$ where $\\lambda_{geo}$ is the canonical Liouville form on $T^*Q$. \nBy Theorem \\ref{ASring} of Abbondandolo and Schwarz the map \n\\begin{equation}\n\\Psi_{AS,q_1} : \\mathrm{H}_*(\\Omega_{q_1}(Q))\\to \\mathrm{HW}(D_g^*Q , L_{q_1})\n\\end{equation} \\color{black}\nis an algebra isomorphism. \nIt is well-known that there is an algebra isomorphism\n\\begin{equation}\n\\Phi: \\mathbb{Z}_2[\\pi_1(Q,q_1)] \\to \\mathrm{H}_0(\\Omega_{q_1}(Q)).\n\\end{equation}\nComposing these two maps we obtain an injective algebra homomorphism \\color{black}\n\\begin{equation}\n\\widetilde{\\Phi}: \\mathbb{Z}_2[\\pi_1(Q,q_1)] \\to \\mathrm{HW}(D_g^*Q , L_{q_1}).\n\\end{equation}\n\nFor a finitely generated group $G$ and a finite set $\\upsigma$ of generators of $G$, let $\\widehat{\\Gamma}_{\\upsigma}(G)$ be the usual exponential growth of the group $G$ with respect to the set $\\upsigma$; see \\cite[Section VI.C]{delaHarpe}. \nTo a finite set $\\upsigma$ of generators of $\\pi_1(Q,q_1)$, we associate the finite set $S \\subset \\mathbb{Z}_2[\\pi_1(Q,q_1)]$ that is formed by the elements of $\\upsigma$ and its inverses.\nIt is immediate to see that \n\\begin{equation}\n \\widehat{\\Gamma}_{\\upsigma}(\\pi_1(Q,q_1)) = {\\Gamma_S^{\\mathrm{alg}}}(\\mathbb{Z}_2[\\pi_1(Q,q_1)]).\n \\end{equation}\n Using that $\\widetilde{\\Phi}$ is injective we obtain\n \\begin{equation}\n \\widehat{\\Gamma}_{\\upsigma}(\\pi_1(Q,q_1)) = {\\Gamma_S^{\\mathrm{alg}}}(\\mathbb{Z}_2[\\pi_1(Q,q_1)]) \\leq {\\Gamma_{\\widetilde{\\Phi}(S)}^{\\mathrm{alg}}}( \\mathrm{HW}(D_g^*Q , L_{q_1})),\n\\end{equation}\nWe have shown the following \n\\begin{lem} \\label{fundalgrowth}\nIt $\\pi_1(Q,q_1)$ has exponential growth then there exists a finite set $S \\subset \\mathrm{HW}(D_g^*Q , L_{q_1})$ such that $ {\\Gamma_S^{\\mathrm{alg}}}(D_g^*Q , L_{q_1}) >0$.\n\\end{lem}\n\n\n \\subsection{ Proof of statement (A) of Theorem \\ref{spheres} and statement $\\clubsuit$ of Theorem \\ref{maincorollary} } \\\n \n \\textit{Proof of statement (A) of Theorem \\ref{spheres} }\n \n \\color{black}\n Let $G$ be a finitely presented group such that \\color{black}\n \\begin{itemize}\n \\item $\\mathrm{H}_1(G) = \\mathrm{H}_2(G) =0$,\n \\item $G$ has exponential growth,\n \\item $G$ admits a presentation on which the number of relations does not exceed the number of generators.\n \\end{itemize}\n Then, it follows from \\cite{Kervaire}, that for every $n\\geq 4$ there exists a manifold $Q^n$ which is an integral homology sphere and which satisfies $\\pi_1(Q^n) = G$. We denote by $\\varrho(G)$ the minimal number of generators of $G$.\n \n We denote by $D^*Q^n$ the unit disk bundle of $Q^n$, with respect to a Riemannian metric $g$ in $Q^n$, endowed with the canonical symplectic and Liouville forms. \\color{black} We choose a point $q\\in Q^n$ and $g$ generically so that $q$ is not conjugate to itself. Let $S^*Q^n= \\partial D^*Q^n $ be the unit cotangent bundle of $Q^n$. In order to prove our result we consider two distinct cases. \\color{black}\n \n \n \\textbf{Case 1: $n$ is odd and $\\geq 5$.} \\\\\n In this case the Euler characteristic of $Q^n$ vanishes. Because $G$ grows exponentially, we know that $\\mathrm{HW}_0(D^*Q^n,L_q)$ has exponential algebraic growth. Let $N^1$ be the plumbing of $D^*Q^n$ and $D^* S^n$ performed far from $L_q$. \\color{black}\nBy Proposition \\ref{prop:operations}, $\\mathrm{HW}_0(N^1,L_q)$ has exponential algebraic growth. \\color{black}\n \n It is a result of Milnor that the boundary of the plumbing of the unit disk bundles of two odd-dimensional homology spheres of dimension $\\geq 3$ is a homology sphere; see \\cite[Chapter VI - Section 18]{Bredon}. Applying this to the pair\n $D^*Q^n$ and $D^* S^n$ we conclude that $\\partial N^1 $ is a homology sphere. \n Since $N^1$ retracts to the one point union of $Q$ and $S^n$ we know that the homology of $N^1$ \\color{black} is zero in every degree different from $0$ and $n$, where we have $\\mathrm{H}_0(N^1) = \\mathbb{Z}$ and $\\mathrm{H}_n(N^1) = \\mathbb{Z} \\oplus \\mathbb{Z}$. \\color{black}\n \n \n \n \n \n \\textbf{Case 2: $n$ is even and $\\geq 4$.} \\\\\n In this case the Euler characteristic of $Q^n$ is $2$. We consider the plumbing associated to the E8 tree; see \\cite[Chapter VI - Section 18]{Bredon}. To each vertex of the E8 tree we associate a disk bundle in the following way: \n \\begin{itemize}\n \\item to the leftmost vertex we associate $D^*Q^n$,\n \\item to every other vertex we associate $D^* S^n$.\n \\end{itemize}\n We let $N^1$ be the plumbing associated to the E8 tree determined by this choice of disk bundles at each vertex, and assume that the plumbing is done away from a cotangent fibre $L_q\\subset D^*Q^n$ . It was shown by Milnor (see \\cite[Chapter VI - Section 18]{Bredon} ) that $\\partial N^1$ is a homology sphere. Since $N^1$ retracts to the wedge sum of $Q$ and seven copies of $S^n$ determined by the E8 tree, we know that the homology of $N^1$ is zero \\color{black} in every degree different from $0$ and $n$, where we have $\\mathrm{H}_0(N^1) = \\mathbb{Z}$ and $\\mathrm{H}_n(N^1) = \\oplus_{i=1}^8 \\mathbb{Z} $. \\color{black} By Proposition \\ref{prop:operations}, $\\mathrm{HW}_0(N^1,L_q)$ has exponential algebraic growth. \\color{black}\n\\color{black}\n \n\\textbf{We now treat both cases simultaneously.} \\color{black}\nBy attaching $2$-handles to $N^1$ away from $L_q$ we can obtain a simply connected Liouville domain $N^2$ such that $\\mathrm{HW}(N^2,L_q)$ has exponential algebraic growth. We choose the framing of these handle attachments so that the first Chern class of $N^2$ vanishes.\n\n The effect of the handle attachment on the homology of the boundary can be read from the surgery formula in \\cite[Section X.1]{Kosinsky}. \\color{black}\n One concludes that the homology of $\\partial N^2$ coincides with that of $\\partial N_1$ except in degree $2$, and $\\mathrm{H}_2(\\partial N^2)$ is the direct sum of $\\varrho(G)$ copies of $\\mathbb{Z}$. \\color{black}\n\n \nBy Hurewicz' Theorem there is a basis of $\\mathrm{H}_2(\\partial N^2)$ which is composed of embedded $S^2$. Since the first Chern class of $N^2$ vanishes, it follows from \\cite[Lemma 2.19]{McLean2011} that these $S^2$ can be made isotropic and disjoint from $L_q$ by an isotopy and that their symplectic normal bundle is trivial. We can thus perform the Weinstein handle attachment over these spheres. The resulting Liouville domain $N^3$ still contains the Lagrangian $L_q$ and it follows from Proposition \\ref{prop:operations} that $\\mathrm{HW}(N^3,L_q)$ has exponential algebraic growth. By the surgery formula in \\cite[Section X.1]{Kosinsky}, the effect of these handle attachments on the homology of the boundary implies that $\\mathrm{H}_2(\\partial N^3)=0$ and that the homologies of $\\partial N^3$ and $\\partial N^2$ coincide in all other degrees. \\color{black}\n Therefore, $\\partial N^3$ is a simply connected homology sphere. It follows from Whitehead's Theorem for homology \\cite[Corollary 4.33]{Hatcher} that $\\partial N^3$ also has the homotopy groups of a sphere. \\color{black}\n Since the dimension of $\\partial N^3$ is $>5$ the h-cobordism theorem tells us that $\\partial N^3$ is homeomorphic to a sphere. \\color{black}\n Since the smooth spheres under connected sum form a finite group, we can take the connected sum of finitely many copies of $\\partial N^3$ to get the sphere $\\partial N^4$ with the standard smooth structure such that $\\mathrm{HW}(N^4,L_q)$ has exponential algebraic growth. \\color{black}\n This proves statement (A) of Theorem \\ref{spheres}.\n\\qed\n\n\\\n\n\\textit{Proofs of statement $\\clubsuit$ of Theorem \\ref{maincorollary} }\n Let $V$ be a $(2n-1)$-dimensional manifold where $n\\geq 4$, and assume that there exists an exactly fillable contact structure $\\xi$ on $V$. Denote by $M_V$ a Liouville domain whose boundary is $(V,\\xi)$.\n \\color{black}\n Let $N^4$ be the Liouville domain constructed in the proof of statement (A) of Theorem \\ref{spheres}. \nBy Proposition \\ref{prop:operations}, the Liouville domain $N^5=N^4 \\# M_V$ has an asymptotically conical exact Lagrangian $L$ such that $\\mathrm{HW}(N^5,L)$ has exponential algebraic growth. \\color{black}\nThe statement then follows from Theorem \\ref{theorementropy}. \\qed \n\n\\subsection{Proof of statement (B) of Theorem \\ref{spheres} and statement $\\diamondsuit$ of Theorem \\ref{maincorollary}} \n\n\\\n\n\\textit{Proof of statement (B) of Theorem \\ref{spheres}:} \\\\\nWe will consider a carefully chosen $3$-manifold $Q$.\nConsider the Brieskorn manifolds of dimension $3$, $M(p,q,r) = \\{(z_1,z_2,z_3) \\in \\C^3 \\, |\\, {z_1}^p + {z_2}^q + {z_3}^r =0\\} \\cap S^5$. $M(p,q,r)$ is a $\\Z$-homology sphere if $p,q,r$ are relatively prime (see for example \\cite{Saveliev2002}).\nIt was shown by Milnor \\cite{Milnor1975} that its fundamental group $\\pi_1(M(p,q,r))$ is the commutator subgroup of the group \n$G = G(p,q,r) = \\langle \\gamma_1, \\gamma_2, \\gamma_3 \\, | \\, \\gamma_1^p = \\gamma_2^q = \\gamma_3^r = \\gamma_1\\gamma_2\\gamma_3 \\rangle$ , see also \\cite{Seade2006}.\nThe groups $\\Sigma = G\/Z(G)$ are the triangle groups, where $Z(G)$ is the center of $G$. \nConsider the case $p=2$, $q=3$, $r=7$.\n A short computation shows that $G(2,3,7) = [G(2,3,7), G(2,3,7)] = \\pi_1(M(2,3,7))$. One has $\\widehat{\\Gamma}(G(2,3,7)) \\geq \\widehat{\\Gamma}(\\Sigma(2,3,7))$, and the exponential growth of $\\Sigma(2,3,7)$ is $\\log(x)$, where $x \\approx 1.17628$ is equal to Lehmer's Salem number (see \\cite{Hironaka2003} or \\cite{Breuillard2014}). \nWe take $Q=M(2,3,7)$.\nThe integral homology of $D_g^*Q$ is the same as that of $Q$, which is $\\mathbb{Z}$ in degrees $0$ and $3$ and vanishes in all other degrees. Moreover it is clear that $\\pi_1(S^*Q)= \\pi_1(Q\\times S^2)= \\pi_1(Q)$ is generated by the elements $\\gamma_1$ and $\\gamma_2$.\n \\color{black}\n\nLet $N^1$ be the Liouville domain obtained by plumbing $D_g^*Q$ with the unit disk bundle $D^*S^3$ of $S^3$. \n \\color{black} We assume that the plumbing is performed away from the cotangent fibre $L_q$ over a point $q \\in Q$. \nTherefore $L_q$ survives as a conical exact Lagrangian in $N^1$. \nBy Proposition \\ref{prop:operations} we know that $\\mathrm{HW}_*(N^1,L_{q})$ has exponential algebraic growth.\n\nSince $N^1$ is the plumbing of $D_g^*Q$ and $D^*S^3$, and $Q$ and $S^3$ are both homology spheres we obtain that $\\partial N^1$ is a homology sphere; see \\cite[Chapter VI - Section 18]{Bredon}.\n \\color{black}\n Combining this with the fact that $N^1$ retracts to the one point union of $S^3$ and $Q$ we conclude that \\color{black}\n\\begin{itemize}\n\\item $\\mathrm{H}_0(N_1) = \\mathbb{Z}$, $\\mathrm{H}_3(N_1) = \\mathbb{Z} \\oplus \\mathbb{Z}$ and $\\mathrm{H}_i(N_1) =0$ for $i\\neq 0,3$,\n\\item $\\mathrm{H}_0( \\partial N_1) = \\mathbb{Z}$, $\\mathrm{H}_5( \\partial N_1) = \\mathbb{Z}$, and $\\mathrm{H}_i( \\partial N_1) =0$ for $i\\neq 0,5$.\n\\end{itemize}\n\nLet now $\\{\\overline{\\sigma}_1,\\overline{\\sigma}_{2}, \\overline{\\sigma}_3\\}$ be generators of $\\pi_1(\\partial N_1)= \\pi_1(Q)$ corresponding to $\\gamma_1$, $\\gamma_2$ and $\\gamma_3$ respectively. By the h-principle for subcritical isotropic submanifolds of contact manifolds \\cite{GeigesBook} we can isotope the curve $\\overline{\\sigma}_3$ to a curve ${\\sigma}_3$ which is isotropic in $(S_g^*Q,\\xi_{geo})$. We can also assume that ${\\sigma}_3$ does not intersect $\\Lambda_{q} := \\partial L_{q}$. Since ${\\sigma}_3$ is isotropic and has trivial normal bundle we can apply the Weinstein handle attachment \\cite{Weinstein1990} and attach a 2-handle to $ N^1$ over ${\\sigma}_3$, obtaining a new Liouville domain $N^2$. From the presentation of $\\pi_1(Q)$ that we used, it is clear that $\\partial N^2$ is simply connected, and so is $N^2$ by \\cite[Lemma 2.9]{McLean2011}. We choose the framing of the handle attachment so that $\\partial N^2$ is spin. \nUsing the Mayer-Vietoris sequence we obtain that\n$\\mathrm{H}_0(\\partial N^2) = \\mathbb{Z}$, $\\mathrm{H}_2(\\partial N_2) = \\mathbb{Z} $, and $\\mathrm{H}_1(\\partial N_2) =0$. \\color{black}\nBy Smale's classification of spin simply-connected five manifolds \\cite{Smale} it follows that $\\partial N_2$ is $S^3\\times S^2$. \\color{black}\n\n\nSince $N^2$ is obtained from $N^1$ via a subcritical handle attachment and the Lagrangian $L_q$ is far from the attaching locus of this handles, we know that $L_q$ survives as a conical exact Lagrangian in $N^2$. Moreover Proposition \\ref{prop:operations} implies that $\\mathrm{HW}_*(N^2,L_{q})$ has exponential algebraic growth, and it follows from Theorem \\ref{theorementropy} that the contact manifold $\\partial N_2$ has positive entropy. \\qed\n\n\n\\textit{Proof of statement $\\diamondsuit$ of Theorem \\ref{maincorollary}:} \\color{black}\nThe statement is proved by a connected sum argument identical to the one in the proof of statement $\\clubsuit$. \\qed \\color{black}\n\n\\begin{rem} To guarantee the vanishing of the second Stiefel-Whitney class of $\\partial N^2$ one must only guarantee the vanishing of the first Chern class of $N^2$. As observed in the proof of \\cite[Lemma 2.10]{McLean2011}, one can choose the framing when performing the attachments of the 2-handles so as to guarantee the vanishing of the first Chern class of $N^2$.\n \\end{rem}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction\\label{intro}}\n\n\\subsection{Vertical structure of stellar discs}\\label{vertstruct}\n\nThe vertical structure of the stellar disc in late-type galaxies is mainly \ndetermined\nby the star formation history combined with the dynamical heating\n$\\sigma(t)$ of the stellar population due to gravitational\nperturbations. Young stellar subpopulations are confined tightly to the\nmidplane with small velocity dispersion, whereas the older subpopulations are\ndistributed over larger heights due to their higher velocity dispersion. This\nbasic feature is well-known in the Galaxy (Wielen \\cite{wie77}, Freeman\n\\cite{fre91}, Edvardsson et al. \\cite{edv93} e.g.). Dynamical heating leads\nto an increasing mean age of the population with increasing height above the\nmidplane. This results in an increasing mass-to-light ratio and vertical colour index\ngradients from blue to red. In principle, these colour index gradients should be\ndirectly observable in\nedge-on galaxies but unfortunately, strong dust extinction and reddening\noften dominates the vertical colour index distribution. Therefore detailed models are\nnecessary to extract the intrinsic stellar disc properties from\nmulti-colour photometry.\n\nIn a series of papers van der Kruit \\& Searle (\\cite{vdk81,vdk82a,vdk82b})\nstarted to investigate the vertical and radial properties of the surface-brightness\nprofiles of edge-on galaxies. They determined for a number of nearby galaxies\n(including NGC 5907) the radial scale length, the radial cutoff, and the\nvertical scale length. They recognised that the shape of the vertical \nsurface-brightness profiles can be well fitted by a $sech^2$-model disregarding the\ncentral minimum in the case of strong dust lanes. In recent decades much work\nhas been done on the phenomenological analysis of edge-on galaxies\n (e.g. Guthrie \\cite{gut92}, Pohlen et al. \\cite{poh02}). \n \nIn our Galaxy the stellar subpopulations of the thin disc are not homogeneously\nmixed. The vertical thickness of the subpopulations increases with age \n(Wielen \\& Fuchs \\cite{wie88}). Wainscoat et al. (\\cite{wai89}) observed a\nsimilar vertical structure in the edge-on galaxy IC 2531. For the analysis\n of the intrinsic physical structure of galactic discs \nwe started to investigate the vertical surface-brightness and colour index profiles of edge-on\ngalaxies in order to analyse the age distribution of stellar discs\n(Wielen et al. \\cite{wie92}, Just et al. \\cite{jus96}). We used evolutionary\nstellar population models for the intrinsic light distribution and include dust\nextinction in the radiative transfer for the observable vertical surface-brightness\nprofiles. We found that in edge-on galaxies the vertical colour index gradients above\nthe innermost part, where extinction dominates, are a signature of \nthe changing properties of the stellar population with height above the midplane.\nDe Grijs and Peletier (\\cite{dgr00}) also found from a large sample of\nedge-on galaxies that the stellar\npopulation properties in the discs can be seen in the vertical colour index \ngradients above the extinction features.\n\nFor NGC 5907 and other nearby edge-on galaxies Xilouris et al.\n(\\cite{xil97,xil99}) used an extensive radiative transfer code including\nscattered light to model the two-dimensional light distribution of the stellar\ndiscs. Using exponential intrinsic luminosity distributions different for each\nband (B,V,I in the case of NGC 5907) and an exponential disc for the dust\ncomponent, they determined radial and vertical scale lengths of the star light\nand the properties of the dust component. The resulting relative low dust mass \ncorresponds to a gas-to-dust ratio of 810 (corrected for the Helium contribution). \nIn recent years FIR observations offered a new, more\ndirect way to deduce the dust properties of galaxies. Popescu et al. (\\cite{pop00}) \ndeveloped a sophisticated temperature model of the dust component in edge-on\ngalaxies in order to fit the FIR observations and applied it to NGC 891. \nWith this method Misiriotis et al. (\\cite{mis01}) determined for NGC 5907 a dust\nmass 4 times larger than Xilouris et al. (\\cite{xil99}). For the source of the\nFIR excess they adopted an additional young stellar component in a very thin\ndisc, which is completely hidden by dust extinction in the optical \nand NIR bands.\nRecently Stevens et al. (\\cite{ste05}) used\na two-temperature dust model to fit the IRAS and SCUBA data for a sample of spiral\ngalaxies. For NGC 5907 they determine a dust mass only three times larger \nthan the Xilouris et al. value.\n\n\n\\subsection{Content of this paper}\\label{content}\n\n\n\\begin{figure}[t]\n\\hbox{\\hspace{0cm} \n \\psfig{figure=5701fig1small.ps,width=8.5cm,clip=}}\n\\caption[]{Deep R image of NGC 5907, sum of 4 exposures.\nThe field of view is $15\\times 13.25$ arcmin. N is top, E is left.\n(cf Sect.~\\ref{observ})}\n\\label{figimagr}\n\\end{figure}\n\nIn this paper we present an evolutionary stellar disc model for NGC 5907, \nwhich \nexplains both observational results: 1) The exponential vertical profiles in all\noptical bands are reproduced by the outer stellar disc dominated by the older\nthin disc population and are no longer just a set of independent fit functions in\neach band as in Xilouris et al. (\\cite{xil99}).\n2) The large amount of dust determined from FIR\/submm observations by Misiriotis\net al. (\\cite{mis01}) exceeding the dust component found in Xilouris et al. by a\nfactor of four is necessary in our model to obscure the additional light of the\nyoung stellar subpopulations with low mass-to-light ratio near the midplane of the\ndisc. There is no additional recent star burst required to explain the\nadditional star light needed for the dust heating.\n\nThe basis of our disc model is a vertical cut with a self-consistent detailed\ndistribution of stellar subpopulations. The vertical distribution of the stars\nis determined by the dynamical equilibrium of the stellar subpopulations\ndescribed by the star formation history of the disc \nand by the dynamical heating function \n(measured by the increasing velocity dispersion).\nThe gravitational forces of the gas component and the dark matter halo are\nincluded. \nThe intrinsic stellar light distribution is calculated from population \nsynthesis models of simple stellar populations (SSP). The resulting\n vertical profile is\nthen extended to an exponential disc up to the cutoff radius $R_\\mathrm{max}$ by\nscaling the surface densities with constant scale heights. The intrinsic\nstructure is not changed as a function of radius. We compute the observable\nvertical surface-brightness and colour index profiles by radiative transfer calculations with\nan exponential dust component, where the inclination of the disc \nwith respect to the line of sight is fitted.\n\nThe parameters of the basic vertical profile are chosen to match all constraints \nat some fixed radius $R_0$ of the disc, where the disc is dominating the light\nand the bulge can be neglected. At that radius we fix the relative\ncontribution of the gas and dark matter component \n$Q_\\mathrm{g}=\\Sigma_\\mathrm{g}\/\\Sigma_\\mathrm{tot}$ and \n$Q_\\mathrm{h}=\\Sigma_\\mathrm{h}\/\\Sigma_\\mathrm{tot}$ to the total\nsurface density $\\Sigma_\\mathrm{tot}$ up to a maximum height \n$z_\\mathrm{max}$ above the\nmidplane. We will use\n$R_0=10\\,\\mathrm{kpc}\\approx2\\,R_\\mathrm{s}$ with radial scale length \n$R_\\mathrm{s}$ of the stellar \ndisc, which is near the maximum of \n$\\Sigma_\\mathrm{d}\/\\Sigma_\\mathrm{h}$ in the case of an isothermal halo. \nThe scale height of the gas component relative to the effective scale\nheight of the stars $z_\\mathrm{g}\/z_\\mathrm{0}$ and \n$s_\\mathrm{h}=\\sigma_\\mathrm{h}\/\\sigma_\\mathrm{e}$, the ratio of the velocity \ndispersion of the dark matter halo and the maximum velocity dispersion of the\nstars are also determined at $R_0$.\n\nIn an iterative process the crucial fitting functions $SFR(t_\\mathrm{a}-t)$ and \n $\\sigma(t)$ are optimised together with the global parameters: inclination\n $i$, radial scale length $R_\\mathrm{s}$, effective scale height $z_\\mathrm{s}$, dust\n distribution, and total stellar and dust mass. For comparison with observations\n we use a series of vertical surface-brightness (and colour index) profiles parallel to the\n minor axes in U,B,V,R,I bands, which we obtained from corresponding deep \nphotometric images of\n NGC 5907. The same type of intrinsic disc model was also successfully applied\n to the solar neighbourhood in order to analyse the connection between the local\n velocity distribution \n functions of the main sequence stars from the Catalogue of Nearby Stars \n (CNS4, Jahreiss \\& Wielen \\cite{jah97}) and the\n vertical density profiles of these stars (see Just \\cite{jus01,jus02,jus03}).\n\nIn Sect. \\ref{param1} we collect the basic parameters of NGC 5907, which are\nnecessary for the basic scaling of our disc model to the observational data.\nIn Sect. \\ref{components} we describe the building blocks of our disc\nmodel. \nSect. \\ref{project} gives the iteration procedure with details on the scaling\nmethod, radiative transfer and the multi-colour analysis.\nIn Sect. \\ref{observ} our observations and the derivation of vertical colour index and\nsurface-brightness profiles are presented.\nFig.~\\ref{figimagr} shows a deep R image (cf Section~\\ref{observ} as an example\nof for the high quality of data. \nIn Sect. \\ref{result} we show the final comparison of the profiles, give the\nglobal parameters of our best model and discuss in detail the intrinsic\nstructure of the stellar disc.\nA discussion of some crucial aspects follow in Sect. \\ref{discuss}.\nSect. \\ref{conclusion} contains the concluding remarks.\n\n\\section{Basic parameters of NGC 5907 \\label{param1}}\n\nHere we discuss the basic scaling parameters of NGC 5907, namely distance, \nradial and\nvertical scale length and inclination of the stellar disc, and the masses of \nthe stellar, gaseous and dark matter component.\n\n\\subsection{Distance to NGC 5907\\label{dist}}\n\nThere is some uncertainty in the distance determination of nearby\ngalaxies from the Hubble expansion, because the peculiar motion of the galaxies\nis not negligible. In this work we use a distance of $D=11$\\,Mpc with \n$V_\\mathrm{GSR}=817$\\,km\/s from the RC3 catalogue (de Vaucouleurs et al. \\cite{dev91})\nand a Hubble constant of $H_0=75$\\,km\/s\/Mpc. The resulting scale conversion is\n$1\\arcsec=53.3$\\,pc and the distance modulus is $\\Delta m=-30.2$\\,mag.\n\nAnother distance determination for NGC~5907 using the\nTully-Fisher relation leads to a slightly larger distance of\n $D$=11.6 or 12.0\\,Mpc (Sofue \\cite{sof96,sof97}). In this method the main\n uncertainty arises from the conversion of the edge-on luminosity to a face-on\n luminosity. Zepf et al. (\\cite{zep00}) used a distance of $D=13.5\\pm 2.1$\\,Mpc\n from a combination of the R-band Tully-Fischer relation with a peculiar motion\n model and discussed also (due to the lack of resolved giant stars in the outer\n bulge region from HST observations) the\n possibility of a significantly larger distance.\n\n\n\\subsection{Disc parameters \\label{sdisc}}\n\nFor the outer parts of the disc, where extinction is not significant, \nthe brightness distribution of the disc of NGC 5907 can be modelled by\n exponential profiles in the vertical and radial direction. The radial scale\n length increases from 3.7\\,kpc in H-band (Barneby \\& Thronson \\cite{bar94}) to\n more than 5\\,kpc in the blue \n (van der Kruit \\& Searle \\cite{vdk82a}: 5.7\\,kpc in J-band,\nXilouris et al. \\cite{xil99}: 5.02\\,kpc in B-band). \nVan der Kruit \\& Searle also determined the cutoff radius of the exponential\ndisc at $R_\\mathrm{max}=19.3$\\,kpc, which has some influence on the inner parts of the\nradial surface-brightness profiles due to projection effects. \nThe cutoff was not included in the model of Xilouris et al. (\\cite{xil99}) leading \nto an underestimation of the intrinsic scale length.\n\nThe effective scale height of the disc is approximately \nindependent of radius, but depends strongly on the model used, i.e. additional\ncomponents like thick disc, bulge and also on the radial\nprofile for edge-on galaxies. From moderately deep photometry\nvan der Kruit \\& Searle found $z_\\mathrm{s}=410$\\,pc in J\/band and\nXilouris et al. $z_\\mathrm{s}=340$\\,pc in B-band, and Barneby \\& Thronson \n$z_\\mathrm{s}=320$\\,pc in H-band,\nwhich are relatively small values. With a One-disc-model for the very deep \nR-band photometry Morrison et al. (\\cite{mor94}) determined a scale height of\n$z_\\mathrm{s}=467$\\,pc.\n\nFrom the shape of the outer isophotes van der Kruit \\& Searle found an \ninclination of $i=87.0\\degr $, which is widely used as a standard value. Xilouris et\nal. determined independently a value of $i=87.2\\degr $. \nMorrison et al. (\\cite{mor94}) observed a stellar warp in the outer parts of the\ndisc. Therefore the intrinsic inclination may be even\nhigher, since the warp smears out the isophotes\nadditively.\n\nSince in our disc model we assume only one stellar disc for all five observed bands, \nwe redetermine the radial and vertical scale length, the cutoff radius and the\ninclination in the fitting procedure.\n\n \n\\subsection{Masses \\label{mass}}\n\nFor the construction of the disc model we need the relative\nsurface densities of the different components. The scaling to the absolute\nvalues is then a result of the surface-brightness profile fitting \n(cf Sect. \\ref{fitting}), and the total masses of\nthe stellar and gaseous components follow from the radial extrapolation\nto exponential discs. We use the total hydrogen mass $M_\\mathrm{H}$ and convert it to\nthe total gas mass $M_\\mathrm{g}$ which will be used for the determination of the \ngravitational force and to calculate the gas-to-dust ratio $F$. \nFor the dark matter halo we need only a reliable estimate of\nthe local density in the disc.\n\n\\subsubsection{Gas masses \\label{gmass}}\n\nThe atomic gas mass is determined from the 21cm luminosity.\nWe use the standard value\n\\begin{equation}\nM_\\mathrm{HI}=6.9\\times10^9\\rm{M}_\\mathrm{\\rm \\sun}\n\\end{equation}\n (Dumke et al. \\cite{dum97}, Stevens et al. \\cite{ste05}). The molecular\ngas mass is much more uncertain. It is derived by converting CO measurements\nusing the conversion factor $X$, which gives the corresponding $H_2$ mass. \nDumke et al. (\\cite{dum97}) determined individually for the galaxy NGC 5907 \na reduced conversion factor $X$ yielding $M_\\mathrm{H_2}=0.9\\times10^9\\rm{M}_\\mathrm{\\rm \\sun}$. \nThe standard $X$-value would have led to $1.8\\times10^9\\rm{M}_\\mathrm{\\rm \\sun}$, \nwhich demonstrates the intrinsic uncertainty of the molecular gas\nmass determination.\nWe use the molecular gas mass redetermined by \nStevens et al. (\\cite{ste05}) of\n\\begin{equation}\nM_\\mathrm{H_2}=1.7\\times10^9\\rm{M}_\\mathrm{\\rm \\sun}\n\\end{equation}\n (rescaled to the distance of $D=11\\,Mpc$). For the contribution of He to the\ntotal gas mass we add 40\\% to the total hydrogen mass \n$M_\\mathrm{H}=M_\\mathrm{HI}+M_\\mathrm{H_2}$. This leads to the total gas mass of\n\\begin{eqnarray}\nM_\\mathrm{g}&=&1.4\\,M_\\mathrm{H}=1.2\\times10^{10}\\rm{M}_\\mathrm{\\rm \\sun} \\quad\\mbox{with}\\label{eqmgas}\\\\\nM_\\mathrm{H}&=&M_\\mathrm{HI}+M_\\mathrm{H_2}=8.6\\times10^{9}\\rm{M}_\\mathrm{\\rm \\sun} \\quad .\n\\end{eqnarray}\n\nThe radial distribution of HI is flatter and more extended than the \nstellar light distribution and is not exponential. The CO distribution is much\nmore confined to the inner parts than that of HI (Dumke et al. \\cite{dum97}). \nFor simplicity we will also adopt for the scaling of the gas fraction an \nexponential disc, but allow for a scale length larger than the radial scale \nlength of the stellar disc.\nWe will use the same spatial distribution as for the dust component,\nwhich is determined by our fitting procedure.\n\n\\subsubsection{Stellar and dark matter masses \\label{smass}}\n\n\nThe determination of the stellar disc and dark matter (DM) halo mass from the \nrotation curve\nis very uncertain due to the well-known disc-halo degeneracy. \nSackett et al. (\\cite{sac94}) constructed three-component mass models (HI, stellar\ndisc, dark matter halo) to fit the\nHI-rotation curve of Sancisi and van Albada (\\cite{san87}). They found equally\ngood fits with disc mass-to-light ratios of 1, 2, and 4 in the R-band.\nThe models of Sackett et al. with $M\/L=1$ and $M\/L=2$ yield a range of surface\ndensity ratios of stellar disc and halo of\n\\begin{equation}\n\\left.\\frac{\\Sigma_\\mathrm{s}}{\\Sigma_\\mathrm{h}}\\right|_\\mathrm{R_0}=2\\dots 5\n\\quad\\mbox{with}\\quad\nR_0=10\\,\\mathrm{kpc},\\,z_\\mathrm{max}=5.1\\,\\mathrm{kpc}.\n\\end{equation}\nTypical mass-to-light ratios of stellar discs in the V-band are of order unity\nand are dependent on the average age of the population. In the solar\nneighbourhood we have $M\/L_\\mathrm{V}=0.78$ and the extrapolation to the solar cylinder\nis $M\/L_\\mathrm{V}\\approx 1.4$ for the mass-to-light\nratio of the surface density (from CNS4 data, Jahreiss \n\\& Wielen \\cite{jah97}). Our final model yields $M\/L_\\mathrm{V}=0.9$ and $M\/L_\\mathrm{R}=1.0$.\nTherefore we will use the $M\/L_\\mathrm{R}=1$ model of Sackett et al. (\\cite{sac94})\nfor the scaling of the DM halo.\n\nSofue (\\cite{sof96}) presented the joint rotation curve of CO and HI, which shows\nin HI a slightly higher and more pronounced maximum, which is harder to\nreconstruct with a low mass disc of $M\/L_\\mathrm{R}<2$. \nFrom H-band photometry Barneby and Thronson (\\cite{bar94}) investigate mass\nmodels including a flattened bulge and determined a small bulge with scaling\nradius $0.23\\,kpc$ and total mass of $M_\\mathrm{B}=9\\times 10^{9}\\,\\rm{M}_\\mathrm{\\rm \\sun} $.\nThe CO rotation curve of Sofue (\\cite{sof96}) confirms the kink produced by the\nbulge at $V_\\mathrm{c}(R=1\\,$kpc$)\\approx 200$\\,km\/s of model b) in \nBarneby and Thronson (\\cite{bar94}). As a consequence the pollution of the disc\nluminosity with bulge light at radial distances larger than 3\\,kpc is very\nsmall. Therefore we decided to neglect the bulge component in our\nanalysis in order to keep the number of free parameters small.\n \n\n\\subsubsection{Dust mass and extinction\\label{dust}}\n\nOur aim is to construct a disc model with a dust component, which has a mass\ncomparable to \n\\begin{equation}\nM_\\mathrm{d}=6\\times10^7\\rm{M}_\\mathrm{\\rm \\sun}\\,,\n\\end{equation}\nthe value determined by Misiriotis et al. (\\cite{mis01}). \n We use the standard extinction law of Rieke \\&\nLebofsky (\\cite{rie85}), which was confirmed by Xilouris et al. (\\cite{xil99})\nfor NGC 5907 and which also goes into the conversion factor of extinction to dust\nmass.\nIn our model we determine the\nspatial distribution of extinction $A_\\mathrm{V}(R,z)$ (see Eq. \\ref{eqdust}). This\nwill be converted to the dust mass distribution by\n\\begin{equation}\n\\rho_\\mathrm{d}=0.175\\times 10^{-3} A_\\mathrm{V} \\rm{M}_\\mathrm{\\rm \\sun}\\,\\mathrm{pc}^{-3} \\label{eqrhod}\n\\end{equation}\nwith $A_\\mathrm{V}$ in [mag\/kpc]. This is the same conversion factor as used by\nXilouris et al. (\\cite{xil99}) and Misiriotis et al. (\\cite{mis01}).\n\n\n\\section{Building blocks of the disc model \\label{components}}\n\nThe aim of this work is to construct a physical model of the stellar disc\nin order to reproduce the vertical surface-brightness and colour index profiles from U,B,V,R,\nand I-band observations. \nTherefore we pay much attention to the internal structure of the stellar disc.\nThe gas and dust component and the dark matter halo are modelled in a simple way.\nThe bulge contribution is neglected, because the number\nof free parameters and fitting functions would be approximately doubled with\nvery small effect on the parameters of the disc. A thick disc component is\nexcluded by very deep photometry and the faint stellar halo is below our\nobservational limit (Morrison et al \\cite{mor94}).\n We apply our model to those\nregions of the disc where the thin disc strongly dominates.\n\nIn this section we describe the different ingredients necessary to construct the\ndisc model and compute their intrinsic properties. The projection onto the sky and\nthe fitting procedure is given in Sect. \\ref{project}. \n\n\n\\subsection{Self-gravitating disc \\label{grav}}\n\nThe backbone of the disc is a self-gravitating vertical disc profile including \nthe gas component in the thin disc approximation. In this approximation the\nPoisson-Equation is one-dimensional\n\\begin{equation}\n\\frac{\\mbox{\\rm d}^2\\Phi_\\mathrm{self}}{\\mbox{\\rm d} z^2}=4\\pi G \\rho(z) \\quad,\\label{eqpoi}\n\\end{equation}\nwhere $\\rho(z)$ is the self-gravitating density and $\\Phi_\\mathrm{self}(z)$ is the\ncorresponding potential. Since we want to construct the disc in dynamical \nequilibrium, the density of the sub-components will be given as a function \nof the total potential $\\rho_\\mathrm{j}(\\Phi)$ and not of height $z$.\nIn the case of a purely self-gravitating thin disc\n(with $\\Phi_\\mathrm{self}=\\Phi$, i.e. no external potential) the Poisson equation can be\nintegrated leading to\n\\begin{equation}\n\\left(\\frac{\\mbox{\\rm d}\\Phi}{\\mbox{\\rm d} z}\\right)^2\n =8\\pi G \\int_0^{\\Phi}\\rho(\\Phi')\\mbox{\\rm d}\\Phi' \\quad.\\label{eqkz}\n\\end{equation}\nThen the vertical distribution is given by the implicit function $z(\\Phi)$ \nvia direct integration\n\\begin{equation}\nz=\\int_0^{\\Phi}\\mbox{\\rm d}\\Phi'\n \\left[8\\pi G \\int_0^{\\Phi'}\\rho(\\Phi'')\\mbox{\\rm d}\\Phi''\\right]^{-1\/2}\n \\quad.\\label{eqz}\n\\end{equation}\nIf an external potential is included, an iteration process is necessary to\nsolve for the vertical distribution. In order to avoid the iteration at that\npoint, we model all gravitational components by a thin disc approximation.\nWe include in the total\npotential $\\Phi$ the stellar component $\\Phi_\\mathrm{s}$, \nthe gas component $\\Phi_\\mathrm{g}$ and the dark matter halo contribution $\\Phi_\\mathrm{h}$\n\\begin{equation}\n\\Phi(z)=\\Phi_\\mathrm{s}(z)+\\Phi_\\mathrm{g}(z)+\\Phi_\\mathrm{h}(z)\\quad.\\label{eqpot}\n\\end{equation}\nIn order to obtain the force of a\nspherical halo correctly in the thin disc approximation,\n we use a special approximation\n(see subsection \\ref{halo}). The relative\ncontribution of the stellar, the gaseous, and the DM-component to the surface\ndensity (up to $|z|=z_\\mathrm{max}$) are given by the input parameters\n$Q_\\mathrm{s},Q_\\mathrm{g},Q_\\mathrm{h}$.\n\n\n\\subsection{Stellar disc \\label{stars}}\n\nWe assume that the disc of NGC 5907 is, like the thin\ndisc of the Milky Way, composed of a sequence of stellar subpopulations with\nincreasing vertical scale height with age. This can be parametrised by the star\nformation history and the dynamical heating function. For convenience we\nuse here the normalisation to the central profile at $R=0$.\nThe stellar component is composed of a sequence of isothermal subpopulations\ncharacterised by the IMF, the chemical enrichment $[Fe\/H](t)$, \nthe star formation history $SFR(t_\\mathrm{a}-t)$, and the dynamical evolution described by\nthe vertical velocity dispersion $\\sigma(t)$. Here $t$ is the age of the\nsubpopulation running back in time from the present time $t_\\mathrm{a}=12$\\,Gyr\n(which is the adopted age of the disc). \nWe include mass loss due to stellar evolution and retain the\nstellar-dynamical mass fraction $g(t)$ (stars + remnants) only. The mass lost\nby stellar winds, supernovae and planetary nebulae is mixed implicitly to the\ngas component. \n\nWith the Jeans equation the vertical\ndistribution of each isothermal subpopulation is given by\n\\begin{equation}\n\\rho_\\mathrm{s,j}(z)=\\rho_\\mathrm{s0,j}\\exp\\left( \\frac{-\\Phi(z)}{\\sigma^2(t_\\mathrm{j})}\\right)\n\\quad,\n\\end{equation}\nwhere $\\rho_\\mathrm{s,j}$ is actually a 'density rate', the density per age bin. The\nconnection to the $SFR$ is given by the integral over $z$\n\\begin{equation}\ng(t_\\mathrm{j})SFR(t_\\mathrm{a}-t_\\mathrm{j})=\\int_\\mathrm{-\\infty}^{\\infty}\\rho_\\mathrm{s,j}(z)\\mbox{\\rm d} z\\quad.\n\\end{equation}\nThe (half-)thickness $h_\\mathrm{p}(t_\\mathrm{j})$ is defined by\nthe midplane density $\\rho_\\mathrm{s0,j}$ through\n\\begin{equation}\n\\rho_\\mathrm{s0,j}=\\frac{SFR(t_\\mathrm{a}-t_\\mathrm{j})}{2h_\\mathrm{p}(t_\\mathrm{j})}\\quad .\n\\end{equation}\nThe total stellar density is\n\\begin{equation}\n\\rho_\\mathrm{s}(z)=\\int_0^{t_\\mathrm{a}}\\rho_\\mathrm{s,j}(z)\\mbox{\\rm d} t \\quad,\n\\end{equation}\nwhich then determines the potential $\\Phi_\\mathrm{s}(z)$ via\nthe Poisson Eq. (\\ref{eqpoi}). The stellar surface density $\\Sigma_\\mathrm{s}$ is\nconnected to the integrated star formation $S_0$ by the effective\nstellar-dynamical fraction $g_\\mathrm{eff}$\n\\begin{equation}\n\\Sigma_\\mathrm{s}=\\int\\rho_\\mathrm{s}(z)\\mbox{\\rm d} z=g_\\mathrm{eff}S_0\\label{eqSigmas}\n\\end{equation}\nwith\n\\begin{equation}\nS_0=\\int SFR\\, \\mbox{\\rm d} t\\quad \\mbox{and}\\quad \n g_\\mathrm{eff} =\\frac{\\int g(t)SFR(t_\\mathrm{a}-t)\\mbox{\\rm d} t}{S_0}\\quad .\\label{eqS0}\n\\end{equation}\nThe effective scale height $z_\\mathrm{s}$ of the stellar disc is connected to the maximum\nvelocity dispersion $\\sigma_\\mathrm{e}$ of the subpopulations and the total surface\ndensity $\\Sigma_\\mathrm{tot}$ via\n\\begin{equation}\nz_\\mathrm{s}=C_\\mathrm{z}z_\\mathrm{e}=C_\\mathrm{z}\\frac{\\sigma_\\mathrm{e}^2}{2\\pi G \\Sigma_\\mathrm{tot}}\\quad, \\label{eqze}\n\\end{equation}\nwhere $z_\\mathrm{e}$ is the scale height of an isothermal component above a disc\nwith total surface density $\\Sigma_\\mathrm{tot}$. The \nshape correction factor $C_\\mathrm{z}$ is of order unity and is determined at\n $z=(3\\pm 0.5)\\,z_\\mathrm{e}$.\n\nThe metalicity $Z$ affects the stellar lifetimes, luminosities and colours of the\nsubpopulations. $Z$ shows a vertical gradient, since the mean age of the stellar\npopulation is correlated with the vertical distribution.\nIn order to account for the systematic influence of the metal enrichment\nwe adopt a moderate metal enrichment similar to the solar neighbourhood\n(Twarog \\cite{twa80}, Edvardsson et al. \\cite{edv93}). We\nuse the enrichment law for oxygen \n\\begin{equation}\nZ=z_\\mathrm{s}+Y\\ln\\left(1+a\\frac{t}{t_\\mathrm{a}}\\right)\n \\quad\\mbox{with}\\quad Y=\\frac{Z_\\mathrm{p}-z_\\mathrm{s}}{\\ln(1+a)}\n\\end{equation}\nwith $Z$ normalised to the solar abundance and the conversion law\n$[Fe\/H]=2[O\/H]$\nas a simple analytic description to account for the enrichment delay by\nSN1a. This model\nis borrowed from a closed-box model with an $n=2$ Schmidt-law for the star\nformation (Lynden-Bell \\cite{lyn75}, Just et al. \\cite{jus96}).\nWe use $a=5$, initial and present metalicity $z_\\mathrm{s}=0.4$, $Z_\\mathrm{p}=1.13$\ncorresponding to $[Fe\/H]=-0.8$ and $[Fe\/H]=0.1$ with solar metalicity\n$Z_\\mathrm{\\sun}=0.018$, respectively (see Fig. \\ref{figsfr}). \n\nIn the fitting procedure (Sect. \\ref{discparam}) a pair of star formation\nhistory and heating function is selected to derive the intrinsic structure of\nthe disc.\nThe star formation history and heating function of the final model are\nshown in Fig. \\ref{figsfr}.\n\n\\begin{figure}[t]\n\\centerline{\n \\resizebox{0.98\\hsize}{!}{\\includegraphics[angle=270]{5701fg2a.ps}}\n }\n\\centerline{\n \\resizebox{0.98\\hsize}{!}{\\includegraphics[angle=270]{5701fg2b.ps}}\n }\n\\caption[]{\nThe upper panel shows the normalised $SFR\/S_0$ as function of normalised age for the\nfinal model of NGC 5907. The dashed line is the constant SFR of a comparison\nmodel. The lower panel gives the corresponding heating function \n$\\sigma(t)\/\\sigma_\\mathrm{e}$\nnormalised to the final velocity dispersion $\\sigma_\\mathrm{e}$ (full line)\nand the chemical\nenrichment (dotted line). \nThe dashed line is the heating function of the solar neighbourhood\nused for the comparison model.\n}\n\\label{figsfr}\n\\end{figure}\n\n\\subsection{Gas and dust component \\label{gas}}\n\nFor the gas and dust component we use simple models to account for the\ngravitation of the gas and the extinction of the dust. In the radial direction\nwe adopt an exponential profile with scale length \n$R_\\mathrm{d}=q_\\mathrm{d}R_\\mathrm{s}$, where\nwe allow for a difference in the scale length of the stellar component by the\nfactor $q_\\mathrm{d}$.\nFor the extinction\nby the dust component we use a simple exponential profile with vertical\nscale height $z_\\mathrm{d}$. \n\nThe vertical profile of the gas component, which is used for the gravitational \nforce of the gas, is constructed dynamically \nlike the stellar component. The gas distribution is modeled by\ndistributing the gas with a constant rate over the velocity\ndispersion range $\\sigma(t)$ of the young stars up to a maximum age\n$t_\\mathrm{g}$. By varying $t_\\mathrm{g}$ we force the scale height of the gas $z_\\mathrm{g}$\n to the same value as that of the dust component $z_\\mathrm{d}$. \n The surface density of the gas\n is related to the stellar surface density by the ratio \n $Q_\\mathrm{g}\/Q_\\mathrm{s}=\\Sigma_\\mathrm{g}\/\\Sigma_\\mathrm{s}$, which is determined at the reference\n radius $R_0$ in the fitting procedure (cf Sect. \\ref{project}).\n\n\n\\subsection{Dark matter halo \\label{halo}}\n\nThe halo does not fulfil the thin disc approximation. For a spherical halo we\nget the vertical component of the force to lowest order from\n\\begin{equation}\n\\frac{\\mbox{\\rm d}\\Phi_\\mathrm{h}}{\\mbox{\\rm d} z}= \\frac{GM_\\mathrm{R}}{R^2}\\frac{z}{R}\\quad, \n\\end{equation}\nwith $r^2=R^2+z^2$ and $M_\\mathrm{R}$ is the enclosed halo mass inside radius $R$. \nComparing this with the one-dimensional Poisson equation from the thin disc\napproximation (Eq. \\ref{eqpoi} integrated over $z$ \nnear the midplane to lowest order for small $z$)\n\\begin{equation}\n\\frac{\\mbox{\\rm d}\\Phi}{\\mbox{\\rm d} z}= 4\\pi G \\rho_0 z\n\\end{equation}\nwe should use for the local halo density \n\\begin{equation}\n\\rho_\\mathrm{h0}=\\frac{M_\\mathrm{R}}{4\\pi R^3}\n\\end{equation}\nwhich exactly corresponds to the singular\nisothermal sphere. Therefore we can\n use the thin disc approximation also for the halo, if we use the local halo \n density $\\rho_\\mathrm{h0}$ and the halo velocity dispersion \n $\\sigma_\\mathrm{h}$ estimated from the rotation curve\n by adopting an isothermal spherical halo. The effect of\n a cored halo, anisotropy and flattening is neglected here. \n For other halo profiles correction factors would\nbe necessary introducing some inconsistency in the halo profile description and\nleading to a different local halo density. The latter would be more important, \nbecause the effect of the halo potential on the disc is stronger than the\nadiabatic contraction of the halo in the disc potential. \n\n\\begin{figure*}[t]\n\\centerline{\n \\resizebox{0.98\\hsize}{!}{\\includegraphics[angle=0]{5701fig3.eps}}\n }\n\\caption[]{\ncolour index evolution of single stellar populations modelled with the PEGASE code\n(Fioc \\& Rocca-Volmerange \\cite{pegase}). These SSPs are created in a star-burst\nlasting for 25 Myr. We use a Scalo IMF (Scalo \\cite{sca86}). Three different \nmetalicities are shown in the plots: solid: Z = 0.008, long-dashed: \nZ = 0.02, short-dashed: Z = 0.04. \n}\n\\label{colour_evolution}\n\\end{figure*}\n\n\n\\subsection{Radial structure \\label{radial}}\n\nThe parameters of the self-consistent vertical profile described in the last\n section are determined at a scaling radius $R_0=10$\\,kpc. \n The determination of the disc properties at $R_0$ are\n described in Sect. \\ref{discparam}. For the radial structure of the disc we use a\n simple exponential extension up to a cutoff radius $R_\\mathrm{max}$. \n We do not account for the radial variation of intrinsic disc \nproperties, which are 1) the star formation history or metalicity to fit\nthe radial colour index gradients, 2) the increasing dark matter mass fraction\nwith radius, 3) the possible variation of the star\/gas surface density ratio\ndue to different radial scale lengths, 4) the\nbreakdown of the thin disc approximation in the\ninnermost part of the disc, 5) the bulge potential and luminosity in the inner\nregion.\n\nThe stellar disc model is then given by\n\\begin{equation}\n\\rho_\\mathrm{s}(R,z)=\\rho_\\mathrm{s}(z)\\exp(-R\/R_\\mathrm{s}) \\qquad\\mbox{for}\\quad R