Variational principle and Kleinian groups. (Principe variationnel et groupes Kleiniens.) (French) Zbl 1112.37019

Otal, Jean-Pierre; Peigné, Marc

Duke Math. J. 125, No. 1, 15-44 (2004).

Summary: Let \(\Gamma\) be a nonelementary Kleinian group acting on a Cartan-Hadamard manifold \(\widetilde X\); denote by \(\Lambda(\Gamma)\) the nonwandering set of the geodesic flow \((\varphi_t)\) acting on the unit tangent bundle \(T^1(\widetilde X/\Gamma)\). When \(\Gamma\) is convex cocompact (i.e., \(\Lambda(\Gamma)\) is compact), the restriction of \((\varphi_t)\) to \(\Lambda(\Gamma)\) is an Axiom A flow: therefore, by a theorem of Bowen and Ruelle, there exists a unique invariant measure on \(\Lambda(\Gamma)\) which has maximal entropy. In this paper, we study the case of an arbitrary Kleinian group \(\Gamma\). We show that there exists a measure of maximal entropy for the restriction of \((\varphi_t)\) to \(\Lambda(\Gamma)\) if and only if the Patterson-Sullivan measure is finite; furthermore when this measure is finite, it is the unique measure of maximal entropy.
By a theorem of Handel and Kitchens, the supremum of the measure-theoretic entropies equals the infimum of the entropies of the distances \(d\) on \(\Lambda(\Gamma)\); when \(\Gamma\) is geometrically finite, we show that this infimum is achieved by the Riemannian distance \(d\) on \(\Lambda(\Gamma)\).

MSC:

37C40	Smooth ergodic theory, invariant measures for smooth dynamical systems
30F40	Kleinian groups (aspects of compact Riemann surfaces and uniformization)
37D40	Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37B40	Topological entropy
37D35	Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
37F30	Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)

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