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

1996/10/25 by Keith Burns, Gabriel P. Paternain, Burns, Keith +2
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Morphological variations and asymmetry #math.DS

paper · pdf · doi:10.48550/arxiv.math/9610223

arxiv created 1996/10/25 · openalex publication_date 1996/10/25 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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. Mañé showed that the exponential growth rate of the integral of n_T(p,q) over M × M is the topological entropy of the geodesic flow of M. In the present paper we exhibit an open set of metrics on the two-sphere for which the exponential growth rate of n_T(p,q is less than the topological entropy of the geodesic flow for a positive measure set of (p,q)∈ M× M. This answers in the negative questions raised by Mañé.

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