Principe variationnel et groupes Kleiniens

Abstract

Summary -Let Γ be a non-elementary Kleinian group acting on a Cartan-Hadamard manifold X ; denote by Λ(Γ) the non-wandering set of the geodesic flow (φ t ) acting on the unit tangent bundle T 1 (X/Γ). When Γ is convex cocompact (i.e. Λ(Γ) is compact), the restriction of (φ t ) to Λ(Γ) is an Axiom A flow : therefore, by a theorem of Bowen-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 (φ t ) 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-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). Classification A.M.S :Primary 37C40, 37D40, 37B40, , 37D35 Secondary 28A50Dans cet article,X désignera une variété riemannienne complète simplement connexe, dont les courbures sectionnelles sont comprises entre deux constantes négatives −α 2 et −β 2 (0 < α ≤ β). Un groupe Kleinien sera un groupe discret d'isométries deX, non-élémentaire (i.e. qui ne possède pas de sous-groupes abéliens d'indice fini) et sans torsion : un groupe Kleinien Γ opère sans points fixes surX et on note X =X/Γ la variété riemannienne quotient. L'ensemble non-errant du flot géodésique (φ t ) agissant sur le fibré unitaire T 1 X est noté Λ(Γ) : c'est sur cet ensemble que se concentre la dynamique intéressante du flot.Nous allons relier certains invariants de la restriction de (φ t )à Λ(Γ)à l'exposant critique du groupe Γ, un invariant de Γ. Ce nombre δ(Γ) est défini comme l'exposant critique de la série, dite de Poincaré,

Ruelle's inequality in negative curvature

“…As said before, these technical hypotheses are consequence of the assumption on the derivatives of the sectional curvature. In [13] the authors use the regularity of the strong unstable foliation to prove the existence of nice measurable partitions. They follow the ideas in [9] and [6] adapted to the geodesic flow in negative curvature.…”

Escape of mass and entropy for geodesic flows

“…Let pµ n q be a sequence of pg t q-invariant probability measures on T 1 M converging vaguely to 0. By variational principle (see [OP04]), we have lim sup nÑ8 h µn pgq ď δ Γ , so we only need to prove that for every ε ą 0 there exists a sequence pµ n q of pg t qinvariant probability measures such that µ n á 0 and h µn pgq ě h top pgq´ε. Fix ε ą 0 and consider any pg t q-invariant probability measure µ such that h µ pgq ą h top pgq´ε.…”

On the growth of quotients of Kleinian groups

“…This sheds new light on the divergence property of a group. It is now classic that the divergence/convergence of G is of interest when studying the ergodic properties of the geodesic flow (φ t ) t on the unit tangent bundle of the quotient manifold G\X ; namely, the divergence of G is a necessary condition for the existence and unicity of an invariant probability measure of maximal entropy for (φ t ) t [12]. It is also of interest when comparing the critical exponent of a Kleinian group G with that of a subgroup H : if H is divergent and its limit set H is a proper subset of G , then the strict inequality δ G > δ H holds [6].…”