Foliated hyperbolicity and foliations with hyperbolic leaves

Abstract

Given a lamination in a compact space and a laminated vector field $X$ which is hyperbolic when restricted to the leaves of the lamination, we distinguish a class of $X$-invariant probabilities that describe the behaviour of almost every $X$-orbit in every leaf, that we call u-Gibbs states. We apply this to the case of foliations in compact manifolds having leaves with negative curvature, using the foliated hyperbolic vector field on the unit tangent bundle to the foliation generating the leaf geodesics. When the Lyapunov exponents of such an ergodic u-Gibbs states are negative, it is an SRB-measure (having a positive Lebesgue basin of attraction). When the foliation is by hyperbolic leaves, this class of probabilities coincide with the classical harmonic measures introduced by L. Garnett. If furthermore the foliation is transversally conformal and does not admit a transverse invariant measure we show that the ergodic u-Gibbs states are finitely many, supported each in one minimal set of the foliation, have negative Lyapunov exponents and the union of their basins of attraction has full Lebesgue measure. The leaf geodesics emanating from a point have a proportion whose asymptotic statistics is described by each of these ergodic u-Gibbs states, giving rise to continuous visibility functions of the attractors. Reversing time, by considering $-X$, we obtain the existence of the same number of repellors of the foliated geodesic flow having the same harmonic measures as projections to $M$. In the case of only 1 attractor, we obtain a North to South pole dynamics.