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

Keith Burns; Gabriel P. Paternain

doi:10.1017/S0143385797086331

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

Keith Burns^*, Gabriel P. Paternain

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

Let M be a compact C^∞ 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 ≤ T. Mañe showed in [7] that lim _T→∞ 1/T log ∫ _M×M n_T(p,q) dpdq = h_top, where h_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 lim sup _T→∞ 1/T log n_T (p, q) < h_top, for a positive measure set of (p, q) ∈ M × M. This answers in the negative questions raised by Mañé in [7].

Original language	English (US)
Pages (from-to)	1043-1059
Number of pages	17
Journal	Ergodic Theory and Dynamical Systems
Volume	17
Issue number	5
DOIs	https://doi.org/10.1017/S0143385797086331
State	Published - Oct 1997

ASJC Scopus subject areas

Mathematics(all)
Applied Mathematics

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abstract = "Let M be a compact C∞ Riemannian manifold. Given p and q in M and T > 0, define nT(p, q) as the number of geodesic segments joining p and q with length ≤ T. Ma{\~n}e showed in [7] that lim T→∞ 1/T log ∫ M×M nT(p,q) dpdq = htop, where htop 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 lim sup T→∞ 1/T log nT (p, q) < htop, for a positive measure set of (p, q) ∈ M × M. This answers in the negative questions raised by Ma{\~n}{\'e} in [7].",

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AU - Paternain, Gabriel P.

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AB - Let M be a compact C∞ Riemannian manifold. Given p and q in M and T > 0, define nT(p, q) as the number of geodesic segments joining p and q with length ≤ T. Mañe showed in [7] that lim T→∞ 1/T log ∫ M×M nT(p,q) dpdq = htop, where htop 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 lim sup T→∞ 1/T log nT (p, q) < htop, for a positive measure set of (p, q) ∈ M × M. This answers in the negative questions raised by Mañé in [7].

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