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Notes on the Values of Volume Entropy of Graphs DISCRETE AND CONTINUOUS doi:10.3934/dcds.2020221 DYNAMICAL SYSTEMS Volume 40, Number 9, September 2020 pp. 5117{5129 NOTES ON THE VALUES OF VOLUME ENTROPY OF GRAPHS Wooyeon Kim Department of Mathematics University of Wisconsin-Madison Madison, WI 53706, USA Seonhee Lim∗ Department of Mathematical Sciences Seoul National University Seoul, Republic of Korea (Communicated by Yitwah Cheung) Abstract. Volume entropy is an important invariant of metric graphs as well as Riemannian manifolds. In this note, we calculate the change of volume entropy when an edge is added to a metric graph or when a vertex and edges around it are added. In the second part, we estimate the value of the volume entropy which can be used to suggest an algorithm for calculating the persistent volume entropy of graphs. 1. Introduction. Let G be a finite metric graph with more than two vertices. The volume entropy of the graph G is defined as the maximal exponential growth rate of the volume of metric ball B(~x; r) of centerx ~ and radius r in the universal cover of G: log volB(~x; r) h = max lim ; x2V G r!1 r where the volume of B(~x; r) is the sum of length of all the edges (or part of the edges) in B(~x; r) ⊂ Ge. When G is connected, the limit does not depend on the vertex x by subadditivity. Volume entropy, unlike the usual (measure-theoretic) entropy in dynamical sys- tems, is a metric invariant. It is equal to the topological entropy of the geodesic flow [9], [7] for non-positively curved Riemannian manifolds or piecewise Riemmanian manifolds, as well as for finite metric graphs [7], [8]. The volume entropy is an important metric invariant, and it is related to volume entropy rigidity question in Riemannian geometry [2]. See also [7] for a metric graph analogue and [10] for a related question. The pair of topological entropy together with the ergodic period is a complete invariant for the equivalence relation of almost topological conjugacy, in the setting of ergodically supported expansive maps with shadowing property, including Anosov maps [12]. As the volume entropy is equal to the topological 2020 Mathematics Subject Classification. 37D40, 37B40, 92Bxx. Key words and phrases. Volume entropy, persistent entropy of graph, network analysis. This work was supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1601-03 and the National Research Foundation of Korea(NRF) (NRF- 2017R1E1A1A03070779, 2017R1A5A1015626). ∗ Corresponding author: Seonhee Lim. 5117 5118 WOOYEON KIM AND SEONHEE LIM entropy of the time one map of the geodesic flow, which is Anosov, together with the ergodic period, the pair is a complete invariant for the topological conjugacy. Volume entropy is also related to the first Betti number and systole of weighted graphs [1]. Recently, there have been attempts to use the volume entropy as an invariant to distinguish brain networks of certain patients [6]. One way to obtain a local invariant is to remove a vertex and all the edges emanating from it. Another possible approach to use the entropy for networks is to use \Persistent volume entropy" by introducing a parameter " similar to the parameter in persistent homology [4]. Suppose that we are given a metric graph G. For given > 0, delete all the edges of length great than to obtain a graph G" and consider the volume entropy of the graph G". Since the graph G is finite, the set of edge lengths f"1 < ··· < "mg is finite and it is exactly the set of " > 0 such that the graph G"i strictly contains G" for any small enough < i. In the second part of section 2, we suggest an algorithm of calculating entropy by specifying " which will be the threshold for two distinct ways of calculation: Newton's method using characterization in [7] (Theorem 4) and a recursive formula for entropy when an edge is added. Having such applications in mind, in this article, we investigate the change of volume entropy for two types of new graphs obtained from the original graph by either adding an edge or adding a vertex and all the edges emanating from it. In the first part, we consider a graph G and a new graph G0 obtained by adding an edge e. Our main result is the following. Theorem 1.1. Let G be a finite metric graph with two non-adjacent vertices x, y. Let G0 be a graph obtained by attaching an edge e between the vertices x; y of length 0 l0. Then the volume entropy h = hG0 of G satisfies q l0h e = fxx(h)fyy(h) + fxy(h); where fxx; fyy; fxy are the generating functions of G starting and ending at x, start- ing and ending at y, and starting at x and ending at y, respectively. Theorem 1.2. Let G be a finite metric graph with two cycles whose ratio of lengths is Diophantine (see Definition 2.7) and with two non-adjacent vertices x, y. Let G0 be a graph obtained by attaching an edge e between x; y of length l. Let h and h0 be the volume entropy of G and G0, respectively. Then h0 = h + Ce−hl + O(e−(1+γ)hl)) as l ! 1 for some constant C > 0 and any γ < 1. Note that Diophantine condition is generically satisfied by Khintchine Theorem. See Remark 2.10. The proofs of the above theorems use various properties of generating functions. 2. Volume entropy change when adding an edge or a vertex. Let us denote by V G = fv1; ··· ; vkg the vertex set of G and by EG the set of oriented edges of G. 2.1. Basic properties of entropy. We denote an edge from x to y by [x; y]. We denote by B(x; r) a ball of radius r centered at x in G. Definition 2.1. (1) By a path, we mean a concatenation of adjacent edges of consistent orientation, or in other words, a metric path starting and ending at some vertices. For a path p, we denote its length by l(p). NOTES ON THE VALUES OF VOLUME ENTROPY OF GRAPHS 5119 (2) For x; y 2 V G, let us denote by Pxy the set of paths without backtracking [ starting from x and ending at y. Also denote Pxy by Px. y2V G (3) Define the generating function of G from x (from x to y) by X −l(p)t X −l(p)t fx(t) = e (fxy(t) = e ; resp); p2Px p2Pxy where t 2 R: (4) Define Nxy(r) to be the number of paths in G from x to y of length less than r and N(r) to be the number of paths in G from x of length less than r. (5) Define vol(B(~x; r)) to be the sum of length of all the edges in the metric ball B(~x; r) of centerx ~ and radius r in the universal cover G~ of G. The folllowing lemma is a basic observation (see Lemma1 [7] for a proof). Lemma 2.2. For any vertex x 2 V G, the volume entropy h is the exponential log(Nx(r)) growth rate of Nx(r), i.e. h = lim . r!1 r Since the graph G is finite, by summing over all the vertices of G, the volume entropy is also the exponential growth rate of the number of paths of length at most r. There is a trivial upper bound of h: if k + 1 is the maximum of the degree of vertices and l is the minimum of the edge lengths, then the volume of a ball of r=l+1 log k radius r is bounded above by l · k , thus h is bound above by l . The next lemma gives another characterization of the volume entropy using the generating functions. We will prove the following lemma in the more general setting. Let l be a length function on a set P. Denote by N(r) the number of p 2 P satisfying l(p) < r. We say that l is discrete if N(r) is finite for any r > 0. Define the generating function X −l(p)t of l by f(t) = e , where t 2 R. Note that if the length function is defined p2P on a set Px and induced by the metric, then f = fx and N(r) = Nx(r). X Lemma 2.3. Let l be a discrete length function on a set P. Let f(t) = e−l(p)t be p2P log(N(r)) the generating function of l. Then, the exponential growth rate h = lim r!1 r is the infimum of t > 0 for which f(t) converges. Proof. Let h0 be the infimum of t for which f(t) converges. For t > h, 1 1 1 X X X X f(t) = e−l(p)t ≤ N(n + 1)e−nt = en(h−t+o(1)): n=0 p:n≤l(p)<n+1 n=0 n=0 It follows that f(t) converges when t > h, thus we obtain h ≥ h0. On the other hand, if h > h0, then choose t such that h > t > h0. For r > 0; X X f(t)etr = e(r−l(p))t ≥ e(r−l(p))t: p2P fp2Pjl(p)<rg For l(p) < r, we have e(r−l(p))t > 1 thus f(t)etr ≥ N(r). Taking the log and letting r go to infinity, we obtain t ≥ h; a contradiction. Thus h ≤ h0.
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