Ergodicity and topological entropy of  geodesic flows on surfaces

Jan Philipp  Schröder

doi:10.3934/jmd.2015.9.147

Article Contents

2015, Volume 9: 147-167. Doi: 10.3934/jmd.2015.9.147

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Ergodicity and topological entropy of geodesic flows on surfaces

Jan Philipp Schröder^1,

1.
Faculty of Mathematics, Ruhr University Bochum, Universitätsstraße 150, 44780 Bochum

Received: February 2015

Revised: May 2015

Published: August 2015

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Abstract

We consider reversible Finsler metrics on the 2-sphere and the 2-torus, whose geodesic flow has vanishing topological entropy. Following a construction of A. Katok, we discuss examples of Finsler metrics on both surfaces with large ergodic components for the geodesic flow in the unit tangent bundle. On the other hand, using results of J. Franks and M. Handel, we prove that ergodicity and dense orbits cannot occur in the full unit tangent bundle of the 2-sphere, if the Finsler metric has conjugate points along every closed geodesic. In the case of the 2-torus, we show that ergodicity is restricted to strict subsets of tubes between flow-invariant tori in the unit tangent bundle. The analogous result applies to monotone twist maps.

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Mathematics Subject Classification: Primary: 37J35; Secondary: 37E99, 37A25.

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