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We consider a transversally conformal foliation $\mathcal{F}$ of a closedmanifold $M$ endowed with a smooth Riemannian metric whose restriction to eachleaf is negatively curved. We prove that it satisfies the following dichotomy.Either there is a transverse holonomy-invariant measure for $\mathcal{F}$, orthe foliated geodesic flow admits a finite number of physical measures, whichhave negative transverse Lyapunov exponents and whose basin cover a set fullfor the Lebesgue measure. We also give necessary and sufficient conditions forthe foliated geodesic flow to be partially hyperbolic in the case where thefoliation is transverse to a projective circle bundle over a closed hyperbolicsurface.

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