Gibbs u-states for the foliated geodesic flow and transverse invariant measures

Abstract

This paper is devoted to the study of Gibbs u-states for the geodesic flow tangent to a foliation F of a manifold M having negatively curved leaves. By definition, they are the probability measures on the unit tangent bundle to the foliation that are invariant under the foliated geodesic flow and have Lebesgue disintegration in the unstable manifolds of this flow.On the one hand we give sufficient conditions for the existence of transverse invariant measures. In particular we prove that when the foliated geodesic flow has a Gibbs su-state, i.e. an invariant measure with Lebesgue disintegration both in the stable and unstable manifolds, then this measure has to be obtained by combining a transverse invariant measure and the Liouville measure on the leaves.On the other hand we exhibit a bijective correspondence between the set of Gibbs u-states and a set of probability measure on M that we call φ u -harmonic. Such measures have Lebesgue disintegration in the leaves and their local densities have a very specific form: they possess an integral representation analogue to the Poisson representation of harmonic functions. IntroductionThis is the second of a series of three papers in which we study a notion of Gibbs measure for the geodesic flow tangent to the leaves of a closed foliated manifold with negatively curved leaves [2,3].Let F be a foliation of a closed manifold M (we say that (M , F ) is a closed foliated manifold) whose leaves L are endowed with smooth (i.e. of class C ∞ ) Riemannian metrics g L which vary continuously transversally in the smooth topology (we refer to the assignment L → g L as a leafwise metric). The unit tangent bundle of F is the set M of unit vectors tangent to F . An element of M shall be denoted by v or, when we want to specify its basepoint, by (x, v) with x ∈ M and v ∈ T 1 x F , the unit sphere of the tangent space to F at x. The set M is naturally a manifold endowed with a foliation F whose leaves are the unit tangent bundles of the leaves of F (see §3.1). It also carries a flow G t , the foliated geodesic flow, which preserves the leaves of F and whose restriction to a leaf T 1 L is precisely its geodesic flow. We will be mostly interested in negatively curved leafwise metrics i.e. in the case where all metrics g L have negative sectional curvature. In that case the foliated geodesic flow possesses a weak form of hyperbolicity that resembles the classical notion of partial hyperbolicity and that we shall analyze in detail in §3.2. This new notion has been introduced by Bonatti, Gómez-Mont and Martínez in a recent preprint [11]. They called it the foliated hyperbolicity. This means that F admits two continuous subfoliations, called stable and unstable foliations and denoted by W s and W u , which are G t -invariant and respectively uniformly contracted and dilated by G t . One would like to know the extent to which the classical results concerning partial hyperbolicity can be applied to that context.The motivating problem of this paper is the research of SRB measures (or phy...

A Fatou theorem for F-harmonic functions

“…The holonomy inside the leaves could be an obstruction to extend these local densities. But we showed in [Al1,Al2,Al3] that this obstruction is in fact void for a typical leaf. More precisely, let (U i ) i ∈I be a finite covering of M that trivialize the bundle, such that the intersection of any two of the U i 's is connected.…”

“…This terminology is due to Matsumoto [M]. We give in our context a generalization of a theorem of Matsumoto [M] which is very much in the spirit of Theorem E of [Al2]. Theorem 6.2.…”

Gibbs measures for foliated bundles with negatively curved leaves

“…They used Deroin-Kleptsyn's result as well as the bijective correspondence between harmonic measures and some special G t -invariant measures on the unit tangent bundle, that we will be led to introduce later on, proven in [Al2,BMar,Ma]. Then together with Yang, we proved in [AY] the general case using Pesin's theory as well as a criterion for the existence of holonomy-invariant measures proved in [Al3].…”

“…It is possible to lift to the leaves of F • the distributions E ⋆ , ⋆ = s, u, c s, or cu; the lifted distributions are denoted by E ⋆ and are invariant by the flow G t : this is the foliated hyperbolicity [BGM,Al3];…”

Physical measures for the geodesic flow tangent to a transversally conformal foliation

“…First notice that B(µ) is W s -saturated. Using the second item of Lemma 3.4 as well as the absolute continuity of W s inside leaves of F (see for example[4, Theorem 3.7]) it is enough to prove the existence of a Borel set X full for µ such that for every v ∈ X , B(µ) ∩ W cu (v) has full volume in W cu (v).Denote by X 1 ⊂ M the set of points v ∈ M such that Leb cu -almost every point of W cu 1 (v) belongs to B(µ). Since µ is an ergodic Gibbs u-state, X 1 is full for µ.…”