Topological entropy of minimal geodesics and volume growth on surfaces

Let (M,g) be a compact Riemannian manifold of hyperbolic type, i.e M is a manifold admitting another metric of strictly negative curvature. In this paper we study the geodesic flow restricted to the set of geodesics which are minimal on the universal covering. In particular for surfaces we show that the topological entropy of the minimal geodesics coincides with the volume entropy of (M, g) generalizing work of Freire and Mane.


Introduction and main results
Let (M, g) be a compact Riemannian manifold (connected and ∂M = ∅) and p :M → M its universal Riemannian covering, saving π : T M → M for the canonical projection. In [7], Manning introduced the volume entropy (also called volume growth) h(g) of (M, g) defined by h(g) := lim r→+∞ 1 r log volB(p, r), where p ∈M and B(p, r) denotes the open ball with center p and radius r. He proved that this limit exists and is independent of p. Let h top (φ t ) = h top (φ t SM ) denote the topological entropy of the geodesic flow φ t on the unit tangent bundle SM . Manning proved the following estimate: In the case of nonpositive curvature he showed that equality holds. Subsequently this was generalized by Freire and Mañé [2] to metrics without conjugate points. LetM be the closed and φ t -invarint subset of SM consisting of all v ∈ SM such that the geodesic c v withċ v (0) = v is globally minimizing. We denote by M = Dp(M) the projection ofM to SM and by φ t M , φ tM the geodesic flow restricted to M,M, respectively. In [4] Katok and Hasselblatt stated the following theorem, saying that it is enough to consider minimal geodesics to generate exponential complexity (provided h(g) > 0).
Following Klingenberg [5] we call a compact manifold M to be of hyperbolic type, if there exists a metric of strictly negative curvature g 0 on M . We hope to prove an inequality of the kind h top (φ t M ) ≤ h(g), i.e. that equality holds in the above theorem. A first result in this direction is the following. We will introduce the notation h top (φ t , F, β) and the notion of entropy expansiveness in section 2.1. Theorem 1.2. Let (M, g) be a compact Riemannian manifold of hyperbolic type. There is some constant β depending only on (M, g) such that for each compact set K ⊂M we have h top (φ t , π −1 (K) ∩M, β) ≤ h(g).
Using a result of Bowen [1], which we shall prove below in the non-compact setting, we obtain the following. Presently we do not know if φ tM for Riemannian manifolds (M, g) of hyperbolic type of arbitrary dimension is β-entropy-expansive. We shall prove, however, that in the two-dimensional case, β-entropy-expansiveness holds in the non-wandering set of M. This gives the following result.
This paper is organized as follows. In the second section we study topological entropy and local topological entropy for homeomorphisms of metric spaces and following the ideas of Bowen [1] we provide an estimate for the topological entropy. In section 3, we give a complete proof using the ideas provided by Katok and Hasselblatt that the topological entropy of the minimal geodesics is bounded below by the volume growth (theorem 1.1). Moreover, we study topological entropy of minimal geodesics on manifolds of hyperbolic type and give the proof of theorem 1.2. Finally, in section 4 we show that for surfaces the topological entropy of φ t M equals the volume growth of g (theorem 1.4).

Topological Entropy for homeomorphisms of metric spaces
In this section we study discrete dynamical systems. In order to apply our results to geodesic flows φ t , t ∈ R, observe that the topological entropy of φ t defined in the continuous setting coincides with that of the discrete system φ n , n ∈ Z, cf. [4].

Bowen's definition
Here we recall Bowen's definition of topological entropy. Let f : V → V be a homeomorphism of a not necessarily compact metric space (V, d). For each n ∈ N, a metric on V is defined by Let F be a subset of V . We say that a set Y ⊂ V is (n, ε)-spanning for F if the closed ballsB n (y, ε) = {y ∈ V : d n (x, y) ≤ ε}, y ∈ Y cover F . If Y ⊂ F and B n (y, ε) ∩ Y = {y} for all y ∈ Y , we say that Y is an (n, ε)-separated subset of F . Let r n (F, ε) denote the minimal cardinality of (n, ε)-spanning sets for F and let s n (F, ε) denote the maximal cardinality of (n, ε)-separated subsets of F . It is easy to see that for any ε > 0 we have We define the following notions of topological entropy.
Note that for any ε > 0 we have . If we use s n (F, ε) instead of r n (F, ε), we obtain the same value for h top (f, F ). For details on topological entropy we refer to [9]. We need the following less known concept of local entropy introduced by Bowen [1]. For x ∈ V and β > 0 set the β-local entropy of f . We say that f is β-entropy-expansive for β > 0 if h top,loc (f, β) = 0.

An upper bound for the topological entropy of homeomorphisms
In order to make use of the local entropy it will be important to compute entropy on coverings. We consider the following setting. Let (Ṽ ,d) be a metric space and Γ a subgroup of isometries ofṼ acting onṼ . Assume that the quotient V :=Ṽ /Γ is compact and equipped with a metric d such that the projection p :Ṽ → V is a local isometry. Letf :Ṽ →Ṽ be a homeomorphism which commutes with the group Γ and let f : V → V be the projection defined by f (x) = pf p −1 (x) (this is well-defined sincef , Γ commute). f is a homemorphism as well. Recall the following result.
Proposition 2.1 (theorem 8.12 in [9]). For each compact set K ⊂Ṽ we have In particular, if p(K) = V , then We shall prove the following theorem which is a slight extension of a result of Bowen (see [1]). It allows to estimate the topological entropy using coverings and will be crucial for our applications.
Theorem 2.2. Let K ⊂Ṽ be a compact set such that p(K) = V . Then for any The proof of 2.2 rests of the following estimate.
We need the following elementary lemma (see [1]).
Proof of 2.3. In the following fix positive numbers ε, δ, β > 0, a point x ∈ K, an integer n ∈ N and set F :=B n (x, β). We shall try to describe the orbit {x,f x, ...,f n−1 x} by a finite collection of y's in K and their sets Z β (y).
Step 1. (choice of y 1 , ..., y s ∈ K and appropriate neighborhoods V (y i )) By definition of a we find for all y ∈ K some integer m(y) ∈ N and a (m(y), δ/2)- Define the open neighborhoods For N → ∞, R ց β the compact sets decrease to the compact set Z β (y), so we find N (y) ∈ N, R(y) > β, s.th.
The triangle inequality implies that By the compactness of K we find y 1 , ..., Step 2. (describtion of F by the y i 's) We claim the following: Proof of the claim. We find γ, i withf t x ∈ γV (y i ), and hencẽ Step 3. (application of lemma 2.4) As a consequence of ( * * ), the set γE(y i ) is (m(y i ), δ/2)-spanning forf t (F ). We want to apply 2.4, so we define integers 0 = t 0 < ... < t r = n as follows.
(a) If t k ≥ n − n 0 , set r = k + 1 and t r = n.
Eventually we are in case (a) and the process stops. Moreover we have ),B(f tr−1 x, β), respectively of minimal cardinality, so E 0 is also (t 1 − t 0 , δ/2)-spanning for F and E r−1 is also (t r − t r−1 , δ/2)spanning forf tr−1 (F ). Apply 2.4 to We obtain using the definition of m(y i ) and Observe that c depends only on δ, n 0 , β and n 0 in turn is indepenent of x, n.
Now we are able to prove the theorem.

Proof of 2.2.
Let E n be a minimal (n, β)-spanning set for K and let ε, δ > 0. Then K ⊂ x∈EnB n (x, β) and by 2.3 each of the sets in the above union can be (n, δ)-spanned by using only ce (a+ε)n elements where a = h top,loc (f , β). Hence Letting ε, δ → 0, the claim follows using 2.1.
3 Bounds for topological entropy

Lower Bound
We need the following theorem stated in the book [4] of Katok and Hasselblatt on the topological entropy of minimal geodesics on Riemannian manifolds. For the convenience of the reader we will provide here a complete proof of the result, which differs from the one in [4] in small details. Recall the notation p :M → M for the universal cover of M and For the proof of 3.1 we need a lemma similar to lemma 2.4. Recall that s T (A, δ) denotes the maximal cardinality of a (T, δ)-separated subset of A.
Since L is (T, 2δ)-separated, the triangle inequality implies #B(x 1 , . . . , x m ) ≤ 1. Therefore, since the cardinalities of the L i are maximal implying that they are also (t i − t i−1 , δ)-spanning, We have the following: there exists a sequence T k → ∞ such that for otherwise adding up the volume of the annuli B(x, T k + δ(2) \ B(x, T k ) with T k+1 = T k +δ/2 starting at T 0 sufficiently large would yield that the exponential growth rate is less than h · (1 − ε). Let N k be a maximal 2δ-separated set in the annulusB(x, T k +δ/2)\B(x, T k ), For y ∈ N k let c y : [0, d(x, y)] →M be a minimal geodesic segment with c(0) = x and c(d(x, y)) = y. Now, if y 1 , y 2 ∈ N k with y 1 = y 2 we have  In SM the sets S k := Dp(S k ) are (T k , δ/2)-separated. Define the decreasing sequence of compact sets In order to find large separated sets in M we shall find them in the sets M k , observing that for t ∈ [ Assume k is large enough, s.th.
We apply lemma 3.2 and obtain Applying lemma 3.2 again gives where in the last step we assumed that T is large, so that s T (SM, δ/8) ≤ e 2bT . Hence one of the factors in the last product has to be "large", i.e. for some i ∈ {0, ..., m k − 1} we have This proves the theorem:

Upper Bound for manifolds of hyperbolic type
Following Klingenberg [5] we call a compact Riemannian manifold (M, g) of hyperbolic type, if there exists a metric of strictly negative curvature g 0 on M . From now on we assume the existence of such g 0 on the compact Manifold M . When we lift objects such as g, g 0 from M to the universal coverM we will frequently denote them by the same letters. In the following we write d for the metric onM induced by g and d g0 for the one induced by the background metric g 0 . Due to the compactness of M the two metrics onM are equivalent, i.e. there exists a constant C > 0 such that We write d 1 for the metric on SM defined by   In this subsection we prove the following theorem stated in the introduction. As a consequence we immediately obtain corollary 1.3 in the introduction using the results in section 2.2. where SK = π −1 (K). Then there is some constant β such that In order to prove the theorem, we construct spanning sets for F . Let K ⊂M be a compact set with diam K = a. For r > a consider Let K ε , K ε r be minimal ε-spanning sets for K, K r , respectively. For y ∈ K ε , z ∈ K ε r , let α yz : R →M be the g 0 -geodesic connecting y and z such that α yz (0) = y and α yz (d g0 (y, z)) = z. By the Morse lemma, there exists a minimizing ggeodesic c yz : R →M r 0 -close to α yz (R). Set Lemma 3.5. P r is a (r − 1, β)-spanning set for F with respect to the metric d 1 where β is given by β := 5r 0 + (2C 2 + 1)ε.

Proof of 3.4.
We have #K ε r ≤ C ε · volB (x, r + a + ε/2) , C ε := inf y∈M volB(y, ε/2) The two-dimensional case We use the notation introduced at the beginning of section 3.2. Morse [8] studied the structure of minimal geodesics in the universal coverM (called "class A geodesics" there), where M =M /Γ is a closed orientable surface of genus ≥ 2. Apart from the Morse lemma in section 3.2, which is valid in any dimension, the assumption dim M = 2 provides additional information since iñ M the minimizing geodesics intersect quite easily. As a background metric for M we can choose by the uniformisation theorem a metric of constant negative curvature −1 and we use forM the Poincaré model given bỹ This model has a simple boundary at infinity, namelyM (∞) = S 1 . Using the Morse lemma, for pairs ξ − , ξ + ∈ S 1 with ξ − = ξ + we distinguish the minimal g-geodesics lying in bounded distance from the g 0 -geodesic inM joining ξ − , ξ + . Write c(±∞) := lim t→±∞ c(t) = ξ ± (the limit in the euclidean sense in C) M ξ := {ċ(0) | c : R →M is an arc-length g-minimal with c(±∞) = ξ ± }, thenM = ∪ ξ∈BMξ and each classM ξ is non-empty. In the sequel minimal refers to g-minimizing arc-length geodesics c : R →M . (ii) It is easy to see that no two geodesics fromM + ξ (resp.M − ξ ) intersect transversely. We shall refer to this as the graph property ofM ± ξ .

Structure of the minimals
(iii) The setsM ± ξ and henceM 0 and M 0 are closed and φ t -invariant.
By (ii) in 4.2 the setsM 0 ξ have a simple structure inM , so when calculating h top (φ t M ) we would like to stick to M 0 . For this it is important that M 0 is "sufficiently large". Let Ω ⊂ SM denote the non-wandering set of φ t restricted to M. The following proposition is the key observation to obtain h top (φ t M ) = h(g) in the two-dimensional case. Proof. Let v ∈ Dp −1 (Ω) ∩M ξ and U n = B(v, 1/n) ∩M ⊂ SM for n ∈ N. By definition of Ω there exists γ n ∈ Γ− {id} and t n > 0 such that Dγ n φ tn U n ∩U n = ∅. In particular there is some v n ∈ U n such that w n := Dγ n φ tn v n ∈ U n . Assume v / ∈M 0 ξ , so there are two minimals c ± : R →M inM ξ with c ± (0) = c v (t ± ) and c ± (0, ∞) ⊂M ± (v).
Assume now that c vn (∞) = c wn (∞) for all n ∈ N. Interchanging v n , w n and maybe taking a subsequence, we may assume that v n → v and c vn (∞) = ξ + for all n. Moreover we can assume that the c vn (∞) lie in one connected component ofM (∞) − {ξ − , ξ + }, say c vn (∞) ∈M (∞) ∩M + (v). Now, byċ vn (t + ) →ċ v (t + ) and the assumptions on the points at infinity of c vn , c + , there have to be two intersections of c vn , c + for large n, contradicting the minimality of both geodesics.

Entropy in strips of finite width
In this section we will show that the local entropy of the geodesic flow in the non-wandering set Ω ⊂ M 0 of φ t M is vanishing. We work in the universal cover and writeΩ := Dp −1 (Ω) ⊂ SM for the lifted non-wandering set of φ t M . Recall Hence the geodesic flow restricted toΩ is β-entropy-expansive for any β > 0.
Proof. Fix v 0 ∈Ω, β > 0 and some small δ > 0. By 4.3 we find ξ ∈ B with v 0 ∈M 0 ξ and hence Z β (v 0 ) ⊂M 0 ξ . We shall prove that (T − 1, 2δ)-spanning sets E of minimal cardinality for Z β (v 0 ) ∩M + ξ have cardinality growing at most linearly in T . The same arguments work for Z β (v) ∩M − ξ and hence give the proposition. Write The sets A, K T are compact.
Using the minimality of c v , c w one finds Hence with s 0 := s(0) we have Step Proof. Let s 0 be as in step 1 and j ∈ Z, r ∈ [0, δ) with s 0 = jδ + r. Then By definition of A we have d(πv, πw) ≤ 2β and hence again by step 1 Step 3. h top (φ t , A) = 0.
Proof. Consider the family of (oriented) unparametrised curves A is ordered by the graph property ofM + ξ (c < c ′ iff c ′ ⊂M + (c)) and we construct a sequence of geodesics c 1 < ... < c n < ... in A. By closedness of A we find a <-smallest geodesic c 1 in A. If c 1 , ..., c n are already chosen, take c n+1 ∈ A to be the <-smallest geodesic c n+1 > c n , such that the compact segment c n ∩K T is not entirely contained in the open tube T (c n+1 , δ/3). By construction, there is some p n ∈ c n ∩ K T with d(p n , c n+1 (R)) ≥ δ/3, hence the upper open half disc lies in the open strip between c n < c n+1 (for δ small, s.th. c n does not return to D n by minimality). Moreover all half discs D n are contained in a δ/3neighborhood of K T and disjoint, since the c i are ordered. As the volume of D n is bounded from below by some constant C(δ) using standard comparison theorems and the compactness of M , and the volume of the K T -neighborhood is finite, growing linearly with T , the above construction stops at some finite N (T ), again N (T ) growing at most linearly. On the other hand, by construction for any c ∈ A we find some i ∈ {1, ..., N (T )} such that c ∩ K T ⊂ T (c i , δ/3). Choose the parameterization of the c i such that v i :=ċ i (0) ∈ A. Now by step 2 the set is (T − 1, 2δ)-spanning for A w.r.t. d 1 with cardinality #E(T, δ) = N (T ) · #F (v i , δ) = N (T ) · (4(1 + β/δ) + 1).
Hence for any δ > 0 we have and by letting δ → 0, the claim follows.
We now have the following result.

Using 2.2 and 4.4 we find
But since Ω is has full measure w.r.t. any invariant probability on M, we find (cf. 8.6.1 (ii) in [9]) h top (φ t M ) = h top (φ t Ω ) ≤ h(g).