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

TL;DR

This paper investigates the dynamics of geodesic flows on transversally conformal foliations with negatively curved leaves, establishing a dichotomy between invariant measures and physical measures with negative transverse Lyapunov exponents.

Contribution

It introduces a new dichotomy for the foliated geodesic flow, providing conditions for the existence of physical measures and partial hyperbolicity in specific geometric settings.

Findings

Either a transverse holonomy-invariant measure exists or the flow has finitely many physical measures.

Physical measures have negative transverse Lyapunov exponents.

Conditions for partial hyperbolicity are characterized in certain geometric configurations.

Abstract

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 the case where the foliation is transverse to a projective circle bundle over a closed hyperbolic surface.