TY - JOUR

T1 - Counting geodesics on a Riemannian manifold and topological entropy of geodesic flows

AU - Burns, Keith

AU - Paternain, Gabriel P.

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 1997/10

Y1 - 1997/10

N2 - 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].

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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U2 - 10.1017/S0143385797086331

DO - 10.1017/S0143385797086331

M3 - Article

AN - SCOPUS:0039054224

VL - 17

SP - 1043

EP - 1059

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

IS - 5

ER -