Thermodynamic invariants of Anosov flows and rigidity

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 φ be a C^∞ contact Anosov flow. If its Anosov splitting is C² and it is C⁰ orbit equivalent to the geodesic flow of a closed negatively curved Riemannian manifold, then the cohomological pressure and the metric entropy of φ coincide. This result generalizes a result of U. Hamenstädt for geodesic flows.