Topological Entropy for Geodesic Flows

1979 ◽  
Vol 110 (3) ◽  
pp. 567 ◽  
Author(s):  
Anthony Manning
Keyword(s):  
Topological Entropy ◽  
Geodesic Flows

Counting geodesics on a Riemannian manifold and topological entropy of geodesic flows

1997 ◽  
Vol 17 (5) ◽  
pp. 1043-1059 ◽  
Author(s):  
KEITH BURNS ◽  
GABRIEL P. PATERNAIN
Keyword(s):  
Riemannian Manifold ◽  
Topological Entropy ◽  
Geodesic Flow ◽  
Positive Measure ◽  
Geodesic Flows ◽  
Open Set ◽  
Negative Questions

Let $M$ be a compact $C^{\infty}$ Riemannian manifold. Given $p$ and $q$ in $M$ and $T>0$, define $n_{T}(p,q)$ as the number of geodesic segments joining $p$ and $q$ with length $\leq T$. Mañé showed in [7] that \[ \lim_{T\rightarrow \infty}\frac{1}{T}\log \int_{M\times M}n_{T}(p,q)\,dp\,dq = h_{\rm top}, \] where $h_{\rm top}$ denotes the topological entropy of the geodesic flow of $M$.In this paper we exhibit an open set of metrics on the two-sphere for which \[ \limsup_{T\rightarrow\infty}\frac{1}{T}\log n_{T}(p,q)< h_{\rm top}, \] for a positive measure set of $(p,q)\in M\times M$. This answers in the negative questions raised by Mañé in [7].

scholarly journals Positive topological entropy of Reeb flows on spherizations

2011 ◽  
Vol 151 (1) ◽  
pp. 103-128 ◽  
Author(s):  
LEONARDO MACARINI ◽  
FELIX SCHLENK
Keyword(s):  
Topological Entropy ◽  
Energy Surface ◽  
Klein Bottle ◽  
Closed Surface ◽  
Hyperbolic Manifolds ◽  
Geodesic Flows ◽  
Hamiltonian Flow ◽  
Reeb Flows ◽  
Closed Orbits ◽  
Geometric Idea

AbstractLet M be a closed manifold whose based loop space Ω (M) is “complicated”. Examples are rationally hyperbolic manifolds and manifolds whose fundamental group has exponential growth. Consider a hypersurface Σ in T*M which is fiberwise starshaped with respect to the origin. Choose a function H : T*M → ℝ such that Σ is a regular energy surface of H, and let ϕt be the restriction to Σ of the Hamiltonian flow of H.Theorem 1. The topological entropy of ϕt is positive.This result has been known for fiberwise convex Σ by work of Dinaburg, Gromov, Paternain, and Paternain–Petean on geodesic flows. We use the geometric idea and the Floer homological technique from [19], but in addition apply the sandwiching method. Theorem 1 can be reformulated as follows.Theorem 1'. The topological entropy of any Reeb flow on the spherization SM of T*M is positive.For q ∈ M abbreviate Σq = Σ ∩ Tq*M. The following corollary extends results of Morse and Gromov on the number of geodesics between two points.Corollary 1. Given q ∈ M, for almost every q′ ∈ M the number of orbits of the flow ϕt from Σq to Σq′ grows exponentially in time.In the lowest dimension, Theorem 1 yields the existence of many closed, orbits.Corollary 2. Let M be a closed surface different from S2, ℝP2, the torus and the Klein bottle. Then ϕt carries a horseshoe. In particular, the number of geometrically distinct closed orbits grows exponentially in time.

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