We consider a transversally conformal foliation 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 , or the foliated geodesic flow admits a finite number of physical measures, which have negative transverse Lyapunov exponents and whose basin covers 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.
Cite this article
Sébastien Alvarez, Jiagang Yang, Physical measures for the geodesic flow tangent to a transversally conformal foliation. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 1, pp. 27–51DOI 10.1016/J.ANIHPC.2018.03.009