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

Keith Burns*, Gabriel P. Paternain

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

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. 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].

Original languageEnglish (US)
Pages (from-to)1043-1059
Number of pages17
JournalErgodic Theory and Dynamical Systems
Volume17
Issue number5
DOIs
StatePublished - Oct 1997

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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