On quasi-invariant transverse measures for the horospherical foliation of a negatively curved manifold

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

We consider a transversally conformal foliation $\mathcal{F}$ of a closed manifold $M$ endowed with a smooth Riemannian metric whose restriction to each leaf is negatively curved. We prove that it satisfies the following dichotomy. Either there is a transverse holonomy-invariant measure for $\mathcal{F}$, or the foliated geodesic flow admits a finite number of physical measures, which have negative transverse Lyapunov exponents and whose basin cover a set full for the Lebesgue measure. We also give necessary and sufficient conditions for the foliated geodesic flow to be partially hyperbolic in…

Logarithm laws for strong unstable foliations in negative curvature and non-Archimedian Diophantine approximation

Given for instance a finite volume negatively curved Riemannian manifold $M$, we give a precise relation between the logarithmic growth rates of the excursions into cusps neighborhoods of the strong unstable leaves of negatively recurrent unit vectors of $M$ and their linear divergence rates under the geodesic flow. As an application to non-Archimedian Diophantine approximation in positive characteristic, we relate the growth of the orbits of lattices under one-parameter unipotent subgroups of $\GL_2(\wh K)$ with approximation exponents and continued fraction expansions of elements of the fiel…

Harmonic functions on R-covered foliations and group actions on the circle

Let (M, F) be a compact codimension-one foliated manifold whose leaves are equipped with Riemannian metrics, and consider continuous functions on M that are harmonic along the leaves of F . If every such function is constant on leaves we say that (M, F) has the Liouville property. Our main result is that codimension-one foliated bundles over compact negatively curved manifolds satisfy the Liouville property. Related results for R-covered foliations, as well as for discrete group actions and discrete harmonic functions, are also established.

Geometry of foliations and flows I: Almost transverse pseudo-Anosov flows and asymptotic behavior of foliations

Let F be a foliation in a closed 3-manifold with negatively curved fundamental group and suppose that F is almost transverse to a quasigeodesic pseudo-Anosov flow. We show that the leaves of the foliation in the universal cover extend continuously to the sphere at infinity, hence the limit sets are continuous images of the circle. One important corollary is that if F is a Reebless finite depth foliation in a hyperbolic manifold, then it has the continuous extension property. Such finite depth foliations exist whenever the second Betti number is non zero. The result also applies to other classe…

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

This paper is devoted to the study of Gibbs u-states for the geodesic flow tangent to a foliation with negatively curved leaves. On the one hand we give sufficient conditions for the existence of transverse invariant measures. In particular we prove that when this foliated geodesic flow preserves a Gibbs su-state, i.e. a measure with Lebesgue disintegration both in the stable and unstable horospheres, then it has to be obtained by combining a transverse invariant measure and the Liouville measure on the leaves. On the other hand we study in detail the projections of Gibbs u-states along the un…

Variation of the Liouville measure of a hyperbolic surface

For a compact riemannian manifold of negative curvature, the geodesic foliation of its unit tangent bundle is independent of the negatively curved metric, up to Holder bicontinuous homeomorphism. However, the riemannian metric defines a natural transverse measure to this foliation, the Liouville transverse measure, which does depend on the metric. For a surface S, we show that the map which to a hyperbolic metric on S associates its Liouville transverse measure is differentiable, in an appropriate sense. Its tangent map is valued in the space of transverse Holder distributions for the geodesic…

The structure of branching in Anosov flows of 3-manifolds

In this article we study the topological structure of the lifts to the universal of the stable and unstable foliations of $3$-dimensional Anosov flows. In particular we consider the case when these foliations do not have Hausdorff leaf space. We completely determine the structure of the set of non separated leaves from a given leaf in one of these foliations. As a consequence of this suspensions are characterized, up to topological conjugacy, as the only $3$-dimensional Anosov flows without freely homotopic closed orbits. Furthermore the structure of branching is related to the topology of the…

无共轭点且万有覆叠为可视的紧流形测地流的扩张因子

Expansive factors for geodesic flows of compact manifolds without conjugate points and with visibility universal covering

针对无共轭点且万有覆叠为可视的紧流形，研究其测地流的扩张因子性质，证明了保时间扩张因子的拓扑混合等特性，推导了格林丛可积性及最大熵测度的唯一性等结果。

Equilibrium states in negative curvature

With their origin in thermodynamics and symbolic dynamics, Gibbs measures are crucial tools to study the ergodic theory of the geodesic flow on negatively curved manifolds. We develop a framework (through Patterson-Sullivan densities) allowing us to get rid of compactness assumptions on the manifold, and prove many existence, uniqueness and finiteness results of Gibbs measures. We give many applications, to the Variational Principle, the counting and equidistribution of orbit points and periods, the unique ergodicity of the strong unstable foliation and the classification of Gibbs densities on…