Thermodynamic invariants of Anosov flows and rigidity

Yong Fang

doi:10.3934/dcds.2009.24.1185

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2009, Volume 24, Issue 4: 1185-1204. Doi: 10.3934/dcds.2009.24.1185

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Thermodynamic invariants of Anosov flows and rigidity

Yong Fang^1,

1.
Département de Mathématiques, Université de Cergy-Pontoise, avenue Adolphe Chauvin, 95302, Cergy-Pontoise Cedex

Received: May 31, 2008

Revised: December 31, 2008

Published: October 2009

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Abstract

By using a formula relating topological entropy and cohomological pressure, we obtain several rigidity results about contact Anosov flows. For example, we prove the following result: Let $\varphi$ be a $C^\infty$ contact Anosov flow. If its Anosov splitting is $C^2$ and it is $C^0$ orbit equivalent to the geodesic flow of a closed negatively curved Riemannian manifold, then the cohomological pressure and the metric entropy of $\varphi$ coincide. This result generalizes a result of U. Hamenstädt for geodesic flows.

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Mathematics Subject Classification: Primary: 37A35, 34D20; Secondary: 37D35, 37D40.

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