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 , x∈V G r→∞ 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 ε 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 Gε and consider the volume entropy of the graph Gε. Since the graph G is finite, the set of edge lengths
{ε1 < ··· < εm} is finite and it is exactly the set of ε > 0 such that the graph Gεi strictly contains Gε for any small enough < i. In the second part of section 2, we suggest an algorithm of calculating entropy by specifying ε 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 → ∞ 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 = {v1, ··· , vk} 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 ∈ 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. y∈V 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),
p∈Px p∈Pxy where t ∈ 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 ∈ V G, the volume entropy h is the exponential log(Nx(r)) growth rate of Nx(r), i.e. h = lim . r→∞ 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 ∈ 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 ∈ R. Note that if the length function is defined p∈P 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 p∈P log(N(r)) the generating function of l. Then, the exponential growth rate h = lim r→∞ 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, ∞ ∞ ∞ 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) 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. p∈P {p∈P|l(p) Corollary 2.4. If G is connected, the volume entropy of G is the infimum of t > 0 for which the generating function fxy(t) of G converges, for any x, y ∈ V G. Proof. Let hxy be the exponential growth rate of the number of paths from x to y of length at most r. Denoting by Nxy(r) the number of paths from x to y of length log(Nxy(r)) 0 at most r and hxy = lim . Suppose that y is connected to y and the r→∞ r distance between them is l.We have Nxy0 (r − l) ≤ Nxy(r) ≤ Nxy0 (r + l). Again, by taking the exponential growth rate, we have hxy = hxy0 . For a general graph G, it is immediate that the volume entropy is the maximum of the volume entropy of connected components. 2.2. Adding an edge. In this subsection, we investigate how entropy changes when we add an edge to a given graph G. Let G0 be the graph obtained by attaching an edge e = [x, y] of length l0, i.e. V G0 = V G,EG0 = EG ∪ {e, e}, where e = [y, x] in G. We assume that two vertices x and y in G are not adjacent. Theorem 2.5. Let G be a finite metric graph with two non-adjacent vertices x, y. 0 Let G be a graph obtained by attaching an edge e between x, y of length l0. Then 0 the volume entropy h = hG0 of G satisfies q l0h e = fxx(h)fyy(h) + fxy(h), where fxx, fyy, fxy are generating functions of G, from x to x, from y to y, and from x to y, respectively. Proof. Any path in G0 from x to y can be represented as a concatenation b1e1 ··· bn−1enbn of paths bi in G and the new edge ei = e or e. Each path bi belongs to one of Pxx, Pxy, Pyx, Pyy depending on ei−1 and ei. For example, bi ∈ Pyx if ei−1 = ei = e and bi ∈ Pxx if ei−1 = e, ei = e. Note that if bi ∈ Pxx (bi ∈ Pyy), then bi+1 ∈ Pyx ∪ Pyy (bi+1 ∈ Pxy ∪ Pxx, resp.) since they are seperated by e (e, resp.). Therefore, one can subdivide any given path into a concatenation m m Y Y Pk = pke(qki1 e ··· qkik e)rke(skj1e ··· skjk e) i=1 k=1 of paths Pk = pke(qki1 e ··· qkik e)rke(skj1 e ··· skjk e), where we denote paths in Pxx, Pyx, Pyy, Pxy by p, q, r, s, respectively and ik, jk are nonnegative. When ik = 0(jk = 0), q0(s0) is empty path and we have consecutive e’s. Each path Pk is −l(P )t determined by paths in Pxx, Pyx, Pyy, Pxy and the value e k of the path Pk is a summand of −l0t −l0t ik −l0t −l0t jk fxxe (fxye ) fyye (fyxe ) . Qm since fxy = fyx. Thus the original path k=1 Pk is a summand of the m-th term of the geometric series of common ratio NOTES ON THE VALUES OF VOLUME ENTROPY OF GRAPHS 5121 ∞ !2 2 e−l0t −l0t X −l0t i r = fxxfyy e (e fxy) = fxxfyy 1 − e−l0tf i=0 xy ∞ !2 −l0t X −l0t i = fxxfyye (e fxy) . i=0 Therefore, the generating function of G0 from x to y converges when r < 1 and diverges when r > 1. By Lemma 2.4, the volume entropy hG0 is a zero of 2 e−l0t 1 = fxx(t)fyy(t) −l t . 1 − e 0 fxy(t) ∞ −l0t e −l0t X −l0t i Since −l t = e (e fxy) is positive, 1−e 0 fxy (t) i=0 q l0h 0 e G = fxx(hG0 )fyy(hG0 ) + fxy(hG0 ) holds. f(t) Theorem 2.6. Let f be fx or fxy. Then t is the Laplace transform of N(r) (Nx(r) or Nxy(r), respectively), i.e. for t > 0, Z ∞ f(t) = t N(r)e−trdr. 0 Proof. Note that N(r) is a non-decreasing function and N(0) = 0. For any δ > 0, ∞ X X X f(t) = e−l(p)t = e−l(p)t p∈P n=0 nδ Theorem 2.8. Assume G is Diophantine. Let f be either fx or fxy. Then Ct f(t) = (1 + O((t − h)γ )) t − h as t → h+ for any 0 < γ < 1. Here, C is a constant that depends only on the graph G. 5122 WOOYEON KIM AND SEONHEE LIM 1 Proof. For arbitrary 0 < γ < 1, let α = 1−γ − 1 > 0. There exists a function m(r) hr hr α such that N(r) = Ce + m(r)e , where limr→∞ m(r)r = 0 by (2.1). For fixed k, there exists δ such that if 0 < s < δ, then m(r) < r−α for r > sγ−1. From Theorem 2.6, Z ∞ Ct Z ∞ f(t) = t N(r)e−trdr = + t m(r)e−(t−h)dr. 0 t − h 0 For the second part, assume that t − h < δ. We have Z ∞ Z (t−h)γ−1 Z ∞ m(r)e−(t−h)rdr = m(r)e−(t−h)rdr + m(r)e−(t−h)rdr 0 0 (t−h)γ−1 (2.2) Z ∞ ≤ (t − h)γ−1 max{m(r)} + (t − h)−α(γ−1) e−(t−h)rdr r≥0 (t−h)γ−1 (2.3) ≤ (t − h)γ−1 max{m(r)} + (t − h)−α(γ−1)(t − h)−1 (2.4) r≥0 = (max{m(r)} + 1)(t − h)γ−1. (2.5) r≥0 Ct γ−1 Thus |f(t) − t−h | ≤ (maxr≥0{m(r)} + 1)(t − h) for t − h < δ. It implies Ct γ + f(t) = t−h (1 + O((t − h) )) as x → h for each 0 < γ < 1. Corollary 2.9. Let G be a finite metric Diophantine graph with two non-adjacent vertices x, y. Let G0 be a graph obtained by attaching an edge e between x, y of length l to G. 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 → ∞ for some constant C > 0 and any γ < 1. Proof. Let G be a Diophantine graph. From Theorem 2.8 we can write C t f (t) = zw (1 + O((t − h)γ )) zw t − h for (z, w) = (x, x), (x, y), and (y, y). Applying Theorem 2.5 to these equations, we obtain 0 0 q C h elh = f (h0)f (h0) + f (h0) = (1 + O((h0 − h)γ )) xx yy xy h0 − h h and 0 0 h elh(h0 − h) = Ceh−h (1 + O((h0 − h)γ )) = C + O((h0 − h)γ ), h p where C = ( CxxCyy + Cxy)h. Since elh(h0 − h) is bounded when l → ∞, we have elh(h0 − h) = C + O(e−γhl), i.e. h0 = h + Ce−hl + O(e−(1+γ)hl)). Remark 2.10 (Generic Behavior). We remark that the Diophantine condition for metric graphs is generic, since almost every real number is Diophantine by NOTES ON THE VALUES OF VOLUME ENTROPY OF GRAPHS 5123 Khintchine Theorem or direct calculation. Indeed, if we denote by A the set of real numbers in [0, 1] which are not Diophantine, by definition, ∞ q \ [ [ p A = {x ∈ [0, 1]||x − | < αq−β}. q α,β>0 q=1 p=0 For all α, β > 0, and for the Lebesgue measure µ, ∞ X X p µ(A) ≤ µ({x ∈ [0, 1]||x − | < αq−β}) q q=1 0≤p≤q ∞ ∞ (2.6) X X X = 2αq−β = 2α q−(β−1). q=1 0≤p≤q q=1 Taking β > 2 and sufficiently small α > 0, we obtain µ(A) = 0. 2.3. Adding a vertex. In this subsection, we consider a new graph G0 obtained from G by adding a vertex v = v0 and edges {e1, . . . , en} emanating from v0. Denote the terminal vertex of ei by vi. We first assume that the graph G is connected and then will consider the general case at the end of the section. The basic idea is similar to Theorem 2.5. We will split a cycle in Pvv into primitive paths again. Any cycle p in Pvv is of the form m m m−1 ! Y Y Y Pk = eik qikjk ejk = ei1 qikjk ejk eik+1 qimjm ejm (2.7) k=1 k=1 k=1 of primitive paths P = e q e from v to v, where q is a path in P . k ik ikjk jk ikjk vik vjk To exclude backtracking, there is an additional condition that jk 6= ik+1. Let L, M be (n × n) matrices defined by −(li+lj )t Lij(t) = (1 − δij)e ,Mij(t) = fvivj (t), where fvivj is the generating function of G from vi to vj. L is corresponding to the paths ejk eik+1 with jk 6= ik+1 and M is corresponding to the paths qikjk . Let Fv 0 ij be the generating function of G from v and Fv be the generating function of paths starting with the edge ei and ending with the edge ej. From the expression (2.7), the value e−l(p)t of each cycle consisting of m primitive paths is a summand of the (i, j)- coefficient of the matrix u(ML)m−1MuT , where u(t) := (e−l1t, ··· , e−lnt). Thus ij the generating function Fv can be expressed by the (i, j)-coefficient of the matrix m−1 ! X k T X ij u (ML) Mu . Since Fv(t) = Fv (t), Fv(t) diverges if ||M(t)L(t)|| > k=0 1≤i,j≤n 1 and converges if ||M(t)L(t)|| < 1, where ||·|| denotes the spectral radius. By using Lemma 2.3, we obtain the following theorem. 0 Theorem 2.11. The volume entropy hG0 of G satisfies ||M(hG0 )|| · ||L(hG0 )|| = 1, where || · || denotes the spectral radius. By an argument similar to Corollary 2.9, we have the following result. Theorem 2.12. Let G be a finite metric Diophantine graph. Let h and h0 be the volume entropy of G and G0, respectively. Then h0 = h + C||L(h)|| + O(||L(h)||1+γ ) 5124 WOOYEON KIM AND SEONHEE LIM as lmin = min{li} → ∞ for some constant C > 0 and any γ < 1. −(li+lj )t Proof. Note that ||L(t)|| is continuous since (1 − δij)e s, each components of L(t), are continuous. By Theorem 2.8, we can find constant cijs such that c t f = ij (1 + O((t − h)γ ) vivj t − h + as t → h for any 0 < k < 1. Let K be a matrix such that Kij = cij. Then t t ||M(t)|| = ||K + O((t − h)γ )|| = ||K||(1 + O((t − h)γ )) t − h t − h holds. Applying Theorem 2.11, we obtain h0 1 = ||L(h0)|| · ||M(h0)|| = ||K|| · ||L(h0)||(1 + O((h0 − h)γ )), h0 − h and h0 − h ||L(h0)|| h0 = C (1 + O((h0 − h)γ )) = C + O((h0 − h)γ ) ||L(h)|| ||L(h)|| h 0 0 h0−h where C = ||K||h since h → h and ||L(h )|| → ||L(h)|| as lmin → ∞. Since ||L(h)|| h0−h γ is bounded, we have ||L(h)|| = C + O(||L(h)|| ). Remark 2.13. The constant C of Corollary 2.9,(2.1), and Theorem 2.12 are all equal and depends on the constant of Theorem 2.8. We have an upper bound of this constant from the proof of the Theorem 3.4. The constant of equation (2.1) P and Theorem 2.8 have a upper bound n−1 wi . Thus the constants of Corollary n−2 min{wi} P n−1 wi 2.9 and Theorem 2.12 have upper bound h. wis are components of n−2 min{wi} eigenvector of a matrix composed of generating functions. More details for this upper bound can be seen in the next section. 3. Upper and lower bounds. In this section, we provide an upper bound and a lower bound on the quantities fxx(h), fxy(h), fyy(h) that appear in Theorem 2.5. For the exact calculation of volume entropy, we need to count the number of paths or cycles without backtracking. However, the case with backtracking is much simpler, thus in order to convey the core idea of the proof, we first consider the case with backtracking which is of independent interest in relation to random walk. 3.1. Backtracking case. Let us assume throughout this subsection that G is con- nected and |V G| ≥ 3. Let N(r) be the number of cycles of length less than r. By Lemma 2.2 and log(N(r)) Corollary 2.4, the volume entropy h of G0 satisfies h = lim . r→∞ r For a vertex, v ∈ G, denote by Cv the set of cycles with backtracking with initial vertex and terminal vertex both v. Denote by Dv the set of primitive cycles in Cv, i.e. cycles which do not pass v except at the initial and terminal vertices. Then the length spectrum of the Dv i.e. set of the length of cycles in Dv is a discrete set because G is finite. 0 Denote by Nv(r) and Nv(r) the number of cycles in Cv and Dv of length less 0 than r, respectively. Denote by hC, hD the exponential growth rate of Nv(r),Nv(r), respectively. By Lemma 2.3, hC,(hD) is the infimum of t for which the generating function X X f(t) = e−l(p)t (g(t) = e−l(p)t) p∈Cv p∈Dv NOTES ON THE VALUES OF VOLUME ENTROPY OF GRAPHS 5125 of Cv (Dv, respectively) converges. Then we have a formal form f(t) = g(t) + g(t)2 + g(t)3 + ··· . By Corollary 2.4, hC satisfies X −l(p)hC g(hC) = e = 1. p∈Dv Note that |Dv| > 1 since |V G| ≥ 3 and we have hC > 0. Since g is continuous, there exists > 0 such that g converges in [hC − , hC + ]. Therefore hD < hC. Choose h such that hD < h < hC. Also, for some constants c0 > c > 1, choose h0 such that g(h0) = c0. Then, 0 (hD +o(1))r 0 0 Nv(r) = e and h > h > hD. Let 0 < < h − hD. Theorem 3.1. Let G be a finite metric graph with |V G| ≥ 3 and v ∈ V G. Denote b by Nv (r) the number of cycles with backtracking in G with initial vertex v and b length less than r. Let h be a exponential growth rate of Nv (r). Then there exist m, M, R > 0 which satisfy mehr ≤ N b(r) ≤ Mehr for r > R. Proof. Since g(h0) = c0, we can find R > 0 such that for r > R, 0 (hD +)r Nv(r) < e (3.1) X 0 c < e−h l(p) < c0. (3.2) l(p) {p∈Dv :l(p) X 0 −h(n−1) X (hD +)n −h(n−1) ≤ Nv(n)e < e e n>r n>r e(hD +−h)([r]+1) e(hD +−h)r = ehD + < ehD +. 1 − e(hD +−h) 1 − e(hD +−h) (3.3) For R > 0 chosen above, we choose m, m0 > 0 which satisfy the next two conditions: ehD + m + m0 < e−hR and m = (c − 1)m0. (3.4) 1 − e(hD +−h) hr 0 h0r Lemma 3.2. ∀r ≥ 0,Nv(r) > me + m e − 1. Proof. We use induction. For 0 ≤ r ≤ R, from h0 < h and the above condition (i), hr 0 h0r 0 hR me + m e < (m + m )e < 1 ≤ Nv(r) + 1. hr 0 h0r Now it remains to show that if Nv(r) > me + m e holds in r ∈ [0, r0], also it holds in r ∈ [0, r0 + l1], where l1 is the minimum of the length of cycles in Dv. For a cycle p ∈ Cv such that l(p) ≤ r, we can divide p to pi ∈ Dv to satisfy p = p1 ··· pn, where each path pi have length less than r. For the rest of the proof, 5126 WOOYEON KIM AND SEONHEE LIM let us denote the set {p ∈ Dv : l(p) < r} by {l1 ≤ l2 ≤ · · · ≤ lk}. From counting each case of li with multiplicity, we obtain k X Nv(r) = k + Nv(r − li). (3.5) i=1 If r0 < r ≤ r0 + l1, each r − li is in [0, r0], thus by induction hypothesis, 0 h(r−li) 0 h (r−li) me + m e < Nv(r − li) + 1. Thus, k k k 0 X X h(r−li) X 0 h (r−li) Nv(r) = k + Nv(r − li) > me + m e i=1 i=1 i=1 k k X 0 X 0 = mehr e−hli + m0eh r e−h li i=1 i=1 hD + (hD +−h)r e e 0 > mehr 1 − + cm0eh r 1 − e(hD +−h) h 0 0 e = mehr + m0eh r + (c − 1)m0eh r − m e(hD +)r 1 − e(hD +−h) 0 0 0 = mehr + m0eh r + (c − 1)m0(eh r − e(hD +)r) > mehr + m0eh r − 1, where the last equality uses the second condition of Equation (3.4). As for the upper bound, we claim that ∀r ≥ 0, hr Nv(r) ≤ Me − 1, where M = max{2, 3e−hl1 }. We again use the induction. hr In the case of 0 ≤ r < l1, Nv(r) = 0 ≤ Me − 2 since M ≥ 2. In the case of hr −hl1 hr l1 ≤ r < l2, Nv(r) = 1 ≤ Me − 2 since M ≥ 3e . Thus, Nv(r) ≤ Me − 2 holds for r ∈ [0, l2). hr Now we will show that if r0 ≥ l2 and Nv(r) ≤ Me − 1 holds in r ∈ [0, r0), also it holds in r ∈ [0, r0 + l1). Let assume r ∈ [r0, r0 + l1), then k k X X h(r−li) Nv(r) = k + Nv(r − li) ≤ k + (Me − 2) i=1 i=1 k X ≤ Mehr e−hli − k ≤ Mehr − 2. i=1 Last inequality holds because r ≥ l2 implies k ≥ 2. 3.2. Non-backtracking case. Now, we come back to the more refined calculation of volume entropy and treat the non-backtracking case. Let us assume throughout the section that G is connected and the number of cycles is more than one, so that hG > 0. Denote the edges in G emanating from v0 by e1, ··· , en and the terminal vertex of ei by vi, for i = 1, ··· , n. We may assume n ≥ 3 because removing vertices whose valency is less than 3 doesn’t affect the volume entropy. Denote by Cij the set of cycles starting with the edge ei and ending with the edge ej. Denote by Dij the set of primitive cycles in Cij which do not pass v0 except in the beginning and at the end. Let us denote the set {l(p)|p ∈ Dij, l(p) < r} by NOTES ON THE VALUES OF VOLUME ENTROPY OF GRAPHS 5127 {lij ≤ lij ≤ · · · ≤ lij }. Let l = min{lij|1 ≤ i, j ≤ n} and l = max{lij|1 ≤ 1 2 Nij min 1 max 1 X −l(p)t X −l(p)t i, j ≤ n}. Let fij(t) = e and gij(t) = e . Note that fv0 (t) = p∈Cij p∈Dij X X fij(t) and gv0(t) = gij(t). 1≤i,j≤n 1≤i,j≤n X Let A(t) = (aij(t))1≤i,j≤n be the n by n matrix such that aij(t) = gik(t). k6=j m Note that (A (t))ij is the generating function of cycles starting ei and ending ∞ X X m G without ej, passing v0 m times. Thus fik(t) = ( A (t))ij. Since h is k6=j m=1 a infimum of t for which f(t) converges, the spectral radius ||A(t)|| must be 1 at t = hG. By Perron-Frobenius Theorem, there exists a positive vector w = t (w1, w2, ··· , wn) such that Aw = w and wi > 0 for i = 1, ··· , n. 0 Denote by Nij(r) and Nij(r) the number of cycles in Cij and Dij of length less n X than r, respectively. Let Ni(r) = Nij(r). By similar argument we used to obtain j=1 Equation (3.5), we also obtain n X X X ij Ni(r) = (1 + Nk(r − lm)). j=1 ij k6=j lm 0 (hD +)r Nij(r) < e ∀1 ≤ i, j ≤ n, X −h0l(p) e > cgij(h) ∀1 ≤ i, j ≤ n. {p∈Dij :l(p) X −hl(p) As Equation (3.3), we can find a constant a > 0 such that e {p∈Dij :l(p)≥r−lmax} ≤ ae(hD +−h)r holds for r > R. For R > 0 chosen above, we choose m, m0 > 0 which satisfy the next two condi- tions: n 0 −hR X X 0 (m + m ) max wi < e and am wk = (c − 1)m min wi. (3.6) 1≤i≤n 1≤i≤n j=1 k6=j hr 0 h0r Lemma 3.3. ∀r ≥ lmax,Ni(r) > mwie + m wie (1 ≤ i ≤ n). 0 Proof. We use induction. For lmax ≤ r ≤ R, from h < h and the above condition (i), hr 0 h0r 0 hR mwie + m wie < (m + m )( max wi)e < 1 ≤ Ni(r). 1≤i≤n 5128 WOOYEON KIM AND SEONHEE LIM hr 0 h0r Now it remains to show that if Ni(r) > mvie + m vie holds in r ∈ [lmax, r0], also it holds in r ∈ [lmax, r0 + lmin]. Thus, n X X X h(r−lij ) 0 h0(r−lij ) Ni(r) > (1 + (mwke m + m wke m )) j=1 ij k6=j lm hr By lemma, there exists m > 0 such that N(r) > me for r ≥ lmax. This statement also holds for r ≥ lmin since N(r) ≥ 1 for r ≥ lmin. As for the upper bound, we claim that ∀r ≥ 0, 1 N (r) ≤ Mw ehr − , ∀1 ≤ i ≤ n. i i n − 2 We again use the induction. There exists some constant M > 0 satisfying the above n−1 1 inequality for r ∈ [0, li] for each i. Indeed, take M = n−2 . min {wi} 1≤i≤n hr 1 Now we will show that if r0 ≥ li and Ni(r) ≤ Mwie − n−2 for all 1 ≤ i ≤ n and r ∈ [0, r0], also it holds in r ∈ [0, r0 + lmin]. Let assume r ∈ [r0, r0 + lmin], then by induction hypothesis, n X X X h(r−lij ) 1 N (r) ≤ (1 + (Mw e m − )) (3.7) i k n − 2 j=1 ij k6=j lm The second inequality between (3.8) and (3.9) holds since r > lmax and n X wkaik(h) = wi holds since Aw = w. k=1 hr 1 By induction, Ni(r) ≤ Mwie − n−2 holds for r > 0. Thus upper bound n n P X X n − 1 wi N(r) = N (r) ≤ M w ehr = ehr holds. i i n − 2 min{w } i=1 i=1 i Theorem 3.4. Let G be a finite connected metric graph with more than one cycle. For v ∈ V G, denote by Nv(r) the number of cycles of length less than r which has no backtracking in G and whose initial and terminal vertex are both v. Let h be the volume entropy of the graph G. Let lmin the length of the shortest cycle in G. Then there exist m, M > 0 such that for r ≥ lmin, hr hr me ≤ Nv(r) ≤ Me . Acknowledgments. We would like to thank the referee for the valuable remarks. This work was supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1601-03 and the National Research Foundation of Ko- rea(NRF) (NRF-2017R1E1A1A03070779, 2017R1A5A1015626). REFERENCES [1] F. Balacheff, Volume entropy, weighted girths and stable balls on graphs, Journal of Graph Theory, 55 (2007), 291–305. [2] G. Besson, G. Courtois and S. Gallot, Volume et entropie minimale des espaces localement sym´etriques, Inventiones Mathematicae, 103 (1991), 417–445. [3] A. Broise-Alamichel, J. Parkkonen and F. Paulin, Equidistribution and Counting under Equi- librium States in Negatively Curved Spaces and Graphs of Groups, Progress Mathematics, 329. Birkh¨auser,2019. [4] G. Carlsson, Topology and data, Bull. Amer. Math. Soc. (N.S.), 46 (2009), 255–308. [5] S. Karam, Growth of balls in the universal cover of surfaces and graphs, Trans. Amer. Math. Soc., 367 (2015), 5355–5373. [6] H. Lee, E. Kim, H. Kang, Y. Huh, Y. Lee, S. Lim and D. S. Lee, Volume entropy and information flow in a brain graph, Sci. Rep., 9 (2019), 256. [7] S. Lim, Minimal volume entropy for graphs, Trans. Amer. Math. Soc., 360 (2008), 5089–5100. [8] S. Lim, Entropy rigidity for metric spaces, The Pure and Applied Mathematics of Korea Society of Mathematical Education, 19 (2012), 73–86. [9] A. Manning, Topological entropy for geodesic flows, Annals of Mathematics (2), 110 (1979), 567–573. [10] C. T. McMullen, Entropy and the clique polynomial, Journal of Topology, 8 (2015), 184–212. [11] M. Pollicott, Asymptotic vertex growth for graphs, Spectrum and Dynamics, CRM Proc. Lecture Notes, Amer. Math. Soc., Providence, RI, 52 (2010), 137–145. [12] W. X. Sun, Topological entropy and the complete invariant for expansive maps, Nonlinearity, 13 (2000), 663–673. [13] Z. H. Xia and P. F. Zhang, Exponential growth rate of paths and its connection with dynamics, Progress in Variational Methods, Nankai Ser. Pure Appl. Math. Theoret. Phys., World Sci. Publ., Hackensack, NJ, 7 (2011), 212–224. Received September 2018; 1st revision September 2018; 2nd revision December 2019. E-mail address: [email protected] E-mail address: [email protected]