Topological Entropy for Geodesic Flows Under a Ricci Curvature Condition

Home , Ricci curvature

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 and| injectivity| 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 exponential 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 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

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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 manifold 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

It is known that D(X)=dimX,whenXis a topological manifold.

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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

J 0 = J = γ0 = r = r(J). ∇γ0 ∇J ∇J ∇ ∇∇ Deﬁne A := r =Hessr,sotrA= rand J 0 = AJ. ∇∇ 4

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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]. Then we get trB = trA − = r − 0 0 | | | − t | |4 − t |≤ l0 1 2 2 C(i0,n,k). So 0 B t D1(i0,n,k), for some constant D1. Using the H¨older|| inequality,|| ≤ R l0 l0 l0 1 1 1 2 1 B ( t− 2 ) 2 ( t 2 B ) 2 D(i ,n,k). || || ≤ || || ≤ 0 Z0 Z0 Z0 Then we have I J J 0 J0 (B + )(J) B J + || ||, || || ≤|| || ≤ || t || ≤ || |||| || t l0 J 0 || || D +logl log δ, J ≤ 0 − Zδ || || J (l ) l log(|| || 0 ) D + log( 0 ), J (δ) ≤ δ || || J (δ) J (l ) eDl || || , || || 0 ≤ 0 δ D J (δ) D J (l0) lim e l0 || || = e l0 J 0 (0), || || ≤ δ 0 δ || || → which completes the proof.

t D From Lemma 5.2 in [DSW], we know that J(t) e J(t0) , 0 t t0 < || || ≤ t0 || || ≤ ≤ D J(t) e J(t0) i0. Then J 0(0) = limt 0 || || || ||. || || → t ≤ t0 From lemma 2 and the above inequality, we obtain the following inequality: D D (1) e− J 0(0) t J(t) e J 0(0) t. || || ≤|| || ≤ || || Also we know that

(2) K J(i /4) J 0(0) K J(i /4) , 1|| 0 || ≤ || || ≤ 2|| 0 || for some constants K1,K2depending only on i0,n,k. Consequently, (3) K J(i /4) t J(t) K J(i /4) t. 3|| 0 || ≤|| || ≤ 4|| 0 || If J 0(0),γ0(0) = 0, we can decompose J into tangential and perpendicular componentsh and obtaini6 the above boundedness (3), since tangential component is linear in t [DSW].

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Lemma 3. If h is suﬃciently small, d(φ (v ),φ (w )) d(v ,w ) N(i ,n,k)d(v ,w )h, h p h q − p q ≤ 0 p q for some constant N(i0,n,k).

Proof. Let Q(t, s) be a length minimizing curve from vp to wq , i.e. a piecewise C∞- rectangle which realizes the distance from vp to wq. We may assume the existence of a length minimizing curve. Then we know that 1 i0/4 ∂Q d(v ,w )= dtds p q || ∂S || Z0 Z0 and 1 i0/4+h ∂Q d(φ (v ),φ (w )) dtds. h p h q ≤ || ∂s || Z0 Zh So we get

1 h 1 i0/4+h ∂Q d(φ (v ),φ (w )) d(v ,w ) + dtds. h p h q − p q ≤ || ∂s || Z0 Z0 Z0 Zi0/4 ∂Q ∂Q Let (0,s)=V(s)and (i /4,s)=W(s). Decompose J into J and J ∂s ∂s 0 1 2 such that J (0) = 0,J(i/4) = 0. Then J (0) = V and J (i /4) = W . 1 2 0 || 2 || || || || 1 0 || || || If V = W =0,thenJ(t)=J1(t)=J2(t) = 0 for all t i0/4andvp =wq.So we|| do|| not|| need|| consider this case. From now on we may assume≤ V =0.Then by (3), we have || || 6 (4) K W t J (t) K W t, 3|| || ≤|| 1 || ≤ 4|| || (5) K V t J (i /4 t) K V t. 3|| || ≤|| 2 0 − || ≤ 4|| || Thus we have

h 1 i0/4+h 1 d(φ (v ),φ (w )) d(v ,w ) + J(t) dsdt h p h q − p q ≤ || || Z0 Z0 Zi0/4 Z0 h 1 i0/4+h 1 + J (t) dsdt ≤ || 1 || Z0 Z0 Zi0/4 Z0 h 1 i0/4+h 1 + + J (t) dsdt || 2 || Z0 Z0 Zi0/4 Z0 i0/4+h h 1 ( + K tdt)( V + W ds) ≤ 4 || || || || Zi0/4 Z0 Z0 1 K(i ,n,k)( V + W ds)h, ≤ 0 || || || || Z0 for some constant K(i0,n,k). Now compute the d(vp,wq). Let i K V iK W a = 0 3|| || ,b= 0 3|| || . 4(K V + K W ) 4(K W + K V ) 3|| || 4|| || 3|| || 4|| ||

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On [0,a], we know that J2(t) J1(t) and on [i0/4 b, i0/4], J1(t) J2(t) by (4) and (5). So we obtain|| || ≥ || || − || || ≥ || ||

i0/4 1 d(v ,w )= J dsdt p q || || Z0 Z0 1 a 1 i0/4 J2 J1 dtds + J1 J2 dtds ≥ 0 0 || || − || || 0 i /4 b || || − || || Z Z Z Z 0 − 1 a K V (i /4 t) K W tdtds ≥ 3|| || 0 − − 4|| || Z0 Z0 1 i0/4 + K3 W t K4 V (i0/4 t)dtds. 0 i /4 b || || − || || − Z Z 0 − The integral a K V (i /4 t) K W tdt is the area of 0 3|| || 0 − − 4|| || R (x, y) K W x y K V (i /4 x),x a. { | 4|| || ≤ ≤ 3|| || 0 − ≤} Hence we get a K V 2 J J ai K V /8=i2K /32 3|| || . || 2|| − || 1|| ≥ 0 3|| || 0 3 K W + K V Z0 4|| || 3|| || Similarly we get

i0/4 2 2 K3 W J1 J2 i0K3/32 || || . i /4 b || || − || || ≥ K4 V + K3 W Z 0 − || || || || It is an easy calculation that

i0/4 a ( V 2 W + V W 2)+a ( V 3 + W 3) J dt 1 || || || || || |||| || 2 || || || || , || || ≥ b ( W 2 + V 2)+b V W Z0 1 || || || || 2|| |||| || for some positive constant ai,bi depending only on i0,n,k. Then we obtain 2 2 V + W ( V + W )(b1( V + W )+b2 V W ) || i ||/4 || || || || 2|| || || || 2|| || 3|| |||| ||3 0 J dt ≤ a1( V W + V W )+a2( V + W ) 0 || || || || || || || |||| || || || || || b ( V 3 + W 3)+c ( V 2 W + V W 2) R = 1 || || || || 1 || || || || || |||| || a ( V 3 + W 3)+a ( V 2 W + V W 2) 2 || || || || 1 || || || || || |||| || b V 3 + c V 2 W + c V W 2 + b W 3 = 1|| || 1|| || || || 1|| |||| || 1|| || a V 3 + a V 2 W + a V W 2 + a W 3 2|| || 1|| || || || 1|| |||| || 2|| || b + c W / V + c W 2/ V 2 + b W 3/ V 3 = 1 1|| || || || 1|| || || || 1|| || || || a + a W / V + a W 2/ V 2 + a W 3/ V 3 2 1|| || || || 1|| || || || 2|| || || || C (i ,n,k), ≤ 0 0

for some positive constants c1(i0,n,k),C0(i0,n,k), since we assume V =0and e + e x + e x2 + e x3 || || 6 we know the boundedness of 0 1 2 3 from 2 3 f0 + f1x + f2x + f3x e + e x + e x2 + e x3 e lim 0 1 2 3 = 0 x 0 f + f x + f x2 + f x3 f → 0 1 2 3 0

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and e + e x + e x2 + e x3 e lim 0 1 2 3 = 3 x f + f x + f x2 + f x3 f →∞ 0 1 2 3 3 for positive constants ei,fi and x 0. Consequently, we have ≥ 1 d(φ (v ),φ (w )) d(v ,w ) K( V + W ds)h h p h q − p q ≤ || || || || Z0 1 i0/4 KC ( J dtds)h ≤ 0 || || Z0 Z0 = KC0d(vp,wq)h, which completes the proof. For ﬁxed T ,wehave T/h d(φT(vp),φT(wq)) lim (1 + KC0h) d(vp,wq) ≤ h 0 → eTKC0d(v ,w ). ≤ p q Then by lemma 1, the topological entropy for geodesic ﬂows of compact Riemann- ian manifolds with RicM kand injM i0 is bounded; i.e. h(φ) (2n 1)KC0. This completes the proof≥− of the theorem.≥ ≤ −

References [AC] M. Anderson, J. Cheeger, Cα-compactness for manifolds with Ricci curvature and injectivity radius bounded below, J. Differential Geometry 35 (1992), 265-281. MR 93c:53028 [B] R. Brocks, Abstandsfunktion, Riccikrümmung und Injektivitätsradius, Diplomarbeit, University of Münster (1993). [CE] J. Cheeger, D. G. Ebin, Comparison theorems in Riemannian geometry, Amsterdam, 1975. MR 56:16538 [DSW] X. Dai, Z. Shen, G. Wei, Negative Ricci curvature and isometry group, Duke Math. J. 76 (1994), 59-73. MR 96e:53051 [FM] A. Freire, R. Mañé, On the entropy of geodesic flow in manifolds without conjugate points, Invent. Math. 69 (1982), 375-392. MR 84d:58063 [GHL] S. Gallot, D. Hulin, J. Lafontaine, Riemannian geometry, Springer-Verlag, 1987. MR 88k:53001 [H] J. Heo, Completely multi-positive linear maps and representations of Hilbert C∗- modules,preprint. [KH] A. B. Katok, B. Hasselblatt, Introduction to the modern theory of dynamical systems, Cambridge University Press, 1995. MR 96c:58055 [Ma1] A. Manning, Topological entropy for geodesic flows, Ann. Math. 110 (1979), 567-573. MR 81e:58044 [Ma2] , More topological entropy for geodesic flows, Dynamical systems and turbulence, Warwick (1980), 243-250. MR 84j:58096 [P] S.-H. Paeng, A sphere theorem under a curvature perturbation, Kyushu J. Math. 50 (1996), 459–470.

Department of Mathematics, Seoul National University, Seoul 151-742, Korea E-mail address: [email protected]

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