Ergodicity and topological entropy of  geodesic flows on surfaces

AbstractWe prove the following result: if M is a compact Riemannian surface whose geodesic flow is expansive, then M has no conjugate points. This result and the techniques of E. Ghys imply that all expansive geodesic flows of a compact surface are topologically equivalent.

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Non singular Hamiltonian systems and geodesic flows on surfaces with negative curvature

Publicacions Matemàtiques ◽

10.5565/publmat_42298_01 ◽

1998 ◽

Vol 42 ◽

pp. 267-299 ◽

Cited By ~ 1

Author(s):

E. A. Lacomba ◽

J. G. Reyes

Keyword(s):

Hamiltonian Systems ◽

Negative Curvature ◽

Geodesic Flows ◽

Singular Hamiltonian Systems ◽

Flows On Surfaces

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First derivative of topological entropy for Anosov geodesic flows in the presence of magnetic fields

Nonlinearity ◽

10.1088/0951-7715/10/1/008 ◽

1997 ◽

Vol 10 (1) ◽

pp. 121-131 ◽

Cited By ~ 10

Author(s):

Gabriel P Paternain ◽

Miguel Paternain

Keyword(s):

Magnetic Fields ◽

Topological Entropy ◽

Geodesic Flows ◽

First Derivative

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

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385797086331 ◽

1997 ◽

Vol 17 (5) ◽

pp. 1043-1059 ◽

Cited By ~ 5

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].

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