  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.
