Topological entropyof minimal geodesics  and volume growth on surfaces

Eva Glasmachers; Gerhard Knieper; Carlos Ogouyandjou; Jan Philipp Schröder

doi:10.3934/jmd.2014.8.75

Article Contents

2014, Volume 8, Issue 1: 75-91. Doi: 10.3934/jmd.2014.8.75

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Topological entropy of minimal geodesics and volume growth on surfaces

1.
Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum
2.
Institut de Mathématiques et de Sciences Physiques (IMSP), Université d’Abomey-Calavi 01 BP 613 Porto-Novo

Received: August 2013

Revised: March 2014

Published: June 2014

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Abstract

Let $(M,g)$ be a compact Riemannian manifold of hyperbolic type, i.e $M$ is a manifold admitting another metric of strictly negative curvature. In this paper we study the geodesic flow restricted to the set of geodesics which are minimal on the universal covering. In particular for surfaces we show that the topological entropy of the minimal geodesics coincides with the volume entropy of $(M,g)$ generalizing work of Freire and Mañé.

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Mathematics Subject Classification: Primary: 37A35, 37D40; Secondary: 53D25.

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