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Topological Entropy for Geodesic Flows Under a Ricci Curvature Condition

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Topological Entropy for Geodesic Flows Under a Ricci Curvature Condition PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 6, June 1997, Pages 1873{1879 S 0002-9939(97)03780-5 TOPOLOGICAL ENTROPY FOR GEODESIC FLOWS UNDER A RICCI CURVATURE CONDITION SEONG-HUN PAENG (Communicated by Mary Rees) Abstract. It is known that the topological entropy for the geodesic flow on a Riemannian manifold M is bounded if the absolute value of sectional curvature KM is bounded. We replace this condition by the condition of Ricci curvature andj injectivityj radius. 1. Introduction The topological entropy is the most important invariant related to the orbit growth of a dynamical system. It represents the exponential growth rate for the number of orbit segments. In a sense, topological entropy describes the total expo- nential complexity of the orbit structure. The topological entropy for geodesic flows is closely related to the curvatures of manifolds since geodesic flows depend on the metrics of manifolds. Let φt be a geodesic flow on a Riemannian manifold M and h(φt) be the topological entropy for φt. It was known that if the absolute value of sectional curvature satisfies K <k, then the topological entropy for geodesic flows satisfies h(φ ) (n 1)√k | M | t ≤ − by [Ma2]. Manning also proved that in the case of KM < 0, the topological entropy 1 for geodesic flows is the same as the volume growth rate, limr r− log V (x, r), where V (x, r) is the volume of the ball B(x, r) contained in the→∞ universal covering space [Ma1]. This result was extended to the case of manifolds without conjugate points by Ma˜n´e[FM]. Bishop’s comparison theorem [GHL] implies that if Ricci curvature satisfies Ric kand the injectivity radius of the universal covering space of M is M ≥− infinite, then h(φt) (n 1)k. We prove the following≤ theorem;− p Theorem. Let k, i0 be positive real numbers. Then there exists a constant C(i0,n,k) such that for every n-dimensional compact Riemannian manifold M with RicM k, inj i , the topological entropy for geodesic flow of M is bounded by≥ − M ≥ 0 C(i0,n,k),whereinjM is the injectivity radius of M. Received by the editors August 23, 1995 and, in revised form, October 17, 1995 and December 21, 1995. 1991 Mathematics Subject Classification. Primary 58F17; Secondary 53C20, 53C21, 53C22. Partially supported by the Basic Science Research Institute Program and in part supported by GARC-KOSEF. c 1997 American Mathematical Society 1873 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1874 SEONG-HUN PAENG For the proof, we will use Brocks’ estimate for the Laplacian of the distance function, which will play an important role on estimating the norm of Jacobi fields. We would like to express our gratitude to Professor Hongjong Kim and Professor Dong-Pyo Chi for much kind and helpful advice. 2. Preliminaries Let X be a compact metric space with the distance function d, and let φ = φt : X X be a flow, i.e. 1-parameter subgroup of homeomorphisms. { Define→ } dφ (x, y):=max d(φ(x),φ (y)) 0 t T . T { t t | ≤ ≤ } φ Let Sd(φ, , T ) be the minimal number of balls of radius in the metric dT which cover X. Define 1 hd(φ, ) := lim sup log Sd(φ, , T ). T T →∞ Then the topological entropy for φ is defined as follows: h(φ) := lim hd(φ, ). 0 → The following proposition is Brocks’ estimate on the Laplacian of the distance function. Proposition ([B, DSW]). Let M be an n-dimensional complete Riemannian man- ifold with Ric k,inj i .Letrbe a distance function from p and γ be a M ≥− M ≥ 0 geodesic from p. Consider r and r as functions of t on γ.ThenC1(i0,n,k) n 1 4 ≤ r − C (i ,n,k) for some constants C ,C depending only on i ,n,k on 4 − t ≤ 2 0 1 2 0 [0,i0/2]. The following lemma is important to estimate the upper bound of the topological entropy. It is proved in the discrete time case in [KH]. In this case, almost the same proof can be applied. Lemma 1. Let X be a compact metric space and φ = φt be a flow on M with tC { } d(φt(x),φt(y)) e d(x, y) for some constant C.Thenh(φ) CD(X)where ≤ log b(≤|) | D(X) is the ball dimension defined by D(X) := lim 0 ,andb()is the → log() minimum number of a covering of X by -balls. | | T C Proof. By assumption, d(φt(x),φt(y)) e | |d(x, y), for t T . Then we easily T C ≤ ≤T C know φt(B(x, /e | |)) B(φt(x),), for t T .SoBd(x, /e | |) Bdφ (x, )and ⊂ ≤ ⊂ T T C S(φ, , t) b(/e | |), where Bd(x, r)isther-ball in d-metric. Then we≤ obtain 1 1 T C lim sup log S(φ, , T ) lim sup log b(/e | |) T T ≤ T T →∞ →∞ 1 log b(/eT C ) lim sup | | log(/eT C ) T C | | ≤ T T log(/e | |) | | →∞ D(X) C . ≤ | | It is known that D(X)=dimX,whenXis a topological manifold. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use TOPOLOGICAL ENTROPY 1875 3. Proof of Theorem In the calculation of the entropy, it is an important step to define a metric on the unit tangent bundle, T1M.Letγv(t) be the geodesic on M with γv0 (0) = vp. In Manning’s paper [Ma1], a metric on T1M is defined as follows: d1(vp,wq):= sup d(γv(t),γw(t)). 0 t 1 ≤ ≤ In [KH], they used the following metric: d (v ,w ):=d(p, q)+ P q(v ) w , 2 p q || p p − q|| q where Pp (vp) is the parallel translation of vp along the geodesic from p to q. Now we construct a metric on T M as follows. Since inj i , it is possible to 1 M ≥ 0 identify v T M with the geodesic γ (t), t [0,i /4] with γ0 (0) = v .Thenwe p ∈ 1 v ∈ 0 v p may consider a Jacobi field J along γv(t) as a tangent vector at vp T1M,whereJ ∂Q ∈ ∂Q is generated by a C∞-rectangle Q(t, s) such that (t, 0) = J(t), (t, s) =1 ∂s || ∂t || for any s and 0 t i /4andQ(t, s ) is a geodesic for any fixed s .Inthis ≤ ≤ 0 0 0 identification, we consider a C∞-rectangle, Q(t, s), where 0 s 1and0 t ∂Q ∂Q ≤ ≤ ≤ ≤ i/4, as a C -curve on T M from (0, 0) to (0, 1). 0 ∞ 1 ∂t ∂t Then we define an inner product of tangent vectors and a distance on T1M as follows. Definition. Let J1,J2 be tangent vectors on T1M, i.e. Jacobi fields with above property. Then i0/4 (J ,J )= J (t),J (t) dt, 1 2 h 1 2 i Z0 1 1 i0/4 ∂Q ∂Q 1 ∂Q d(v ,w ) = inf ( , )2 ds =inf dtds, p q ∂s ∂s || ∂s || Z0 Z0 Z0 where the inf is taken over piecewise C∞-curves Q on T1M from vp to wq, i.e. ∂Q ∂Q (0, 0) = v and (0, 1) = w . ∂t p ∂t q Since T1M is a compact Hausdorff space, the above metrics induce the same topology. Now we will prove a key lemma. Lemma 2. Let M be a complete Riemannian manifold with RicM k,injM i0 and let γ(t) be a minimal geodesic starting from q and J(t) is a Jacobi≥− field along≥ D γ such that J(0) = 0 and J 0(0),γ0(0) =0.Then J (t) e t J 0 (0) for some h i || || ≤ || || constant D(i0,n,k) if t<i0/2. Proof. The first half of the proof of this lemma is contained in the proof of the proposition 5.1 of [DSW]. Let v, w TqM and v = w = l0 i0/2. De- ∈ || || ||n ||1 ≤ fine Q(t, s):=exp(tV (s)), where V (s) is the geodesic on S − such that V (0) = ∂Q v, V (s )=w.Letγ(t)=exp tV (0) and r(x)=d(q, x). Then J(t)= (t, 0) is 0 q ∂s a Jacobi field with above property. Then, J 0 = J = γ0 = r = r(J). ∇γ0 ∇J ∇J ∇ ∇∇ Define A := r =Hessr,sotrA= rand J 0 = AJ. ∇∇ 4 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1876 SEONG-HUN PAENG Now we will estimate the A .WriteA(t)=B(t)+I/t. From the Jacobi || || equation J 00 + R(J, T )T =0andJ0 =AJ, we obtain the Riccati equation A0 + 2 A2 + R = 0 and substituting A(t)=B(t)+I/t,wegetB +B2+ B+R=0. 0 t 2 2 2 t 2 t Then, trB0 + B + trB + Ric(γ0) 0since B trB B =trB ,whereB is || || t ≤ || || ≤ the transpose of B and B is a symmetric matrix. 1 Multiplying the above equality by a factor t 2 and integrating along γ(t), we obtain the following equality: l0 l0 l0 1 1 3 1 1 2 2 B t 2 l trB(l ) t− 2 trB(t) t 2 Ric(γ0). || || ≤−0 0 − 2 − Z0 Z0 Z0 n 1 Also using the Proposition ([B, DSW]) in 2, r − C(i,n,k)forsome § |4 − t |≤ 0 n 1 n 1 constant C(i ,n,k)on[0,i /2].
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