Duke Mathematical Journal

Principe variationnel et groupes Kleiniens

Jean-Pierre Otal and Marc Peigné

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Résumé

Soit Γ un groupe Kleinien non élémentaire agissant sur une variété de Cartan-Hadamard $\tilde X$; on note Λ(Γ) l'ensemble non-errant du flot géodésique (φt) agissant sur le fibré unitaire tangent $T1($\tilde{X}$/Γ). Lorsque Γ est convexe cocompact (i.e. Λ(Γ) est compact), la restriction de (φt) à Λ(Γ) est un flot Axiom A : ainsi, d'après un théorème de Bowen-Ruelle, il existe sur Λ(Γ) une unique mesure de probabilité d'entropie maximale, invariante sous l'action de (φt). Dans cet article, nous nous affranchissons de l'hypothèse de compacité de Λ(Γ) et étudions le cas où Γ est quelconque. Nous montrons que la restriction de (φt) à Λ(Γ) admet une mesure de probabilité d'entropie maximale si et seulement si la mesure de Patterson-Sullivan est finie; de plus, lorsque la mesure de Patterson-Sullivan est finie, c'est l'unique mesure d'entropie maximale pour le flot géodésique.

Abstract

Let Γ be a nonelementary Kleinian group acting on a Cartan-Hadamard manifold $\tilde{X}$; denote by Λ(Γ) the nonwandering set of the geodesic flow (φt) acting on the unit tangent bundle $T1($\tilde{X}$/Γ). When Γ is convex cocompact (i.e., Λ(Γ) is compact), the restriction of (φt) to Λ(Γ) is an Axiom A flow: therefore, by a theorem of Bowen and Ruelle, there exists a unique invariant measure on Λ(Γ) which has maximal entropy. In this paper, we study the case of an arbitrary Kleinian group Γ. We show that there exists a measure of maximal entropy for the restriction of(φt) to Λ(Γ) 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 Λ(X); when Γ is geometrically finite, we show that this infimum is achieved by the Riemannian distance d on Λ(X).

Article information

Source
Duke Math. J. Volume 125, Number 1 (2004), 15-44.

Dates
First available in Project Euclid: 25 September 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1096128233

Digital Object Identifier
doi:10.1215/S0012-7094-04-12512-6

Mathematical Reviews number (MathSciNet)
MR2097356

Zentralblatt MATH identifier
1112.37019

Subjects
Primary: 37C40: Smooth ergodic theory, invariant measures [See also 37Dxx] 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 37B40: Topological entropy 37D35: Thermodynamic formalism, variational principles, equilibrium states
Secondary: 28A50: Integration and disintegration of measures

Citation

Otal, Jean-Pierre; Peigné, Marc. Principe variationnel et groupes Kleiniens. Duke Math. J. 125 (2004), no. 1, 15--44. doi:10.1215/S0012-7094-04-12512-6. http://projecteuclid.org/euclid.dmj/1096128233.


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