Asymptotic behavior of foliations

Let $F$ be a foliation which is almost transverse
to a pseudo-Anosov flow $A$ in a closed 3-manifold with
negatively curved fundamental group. Suppose that $A$
is a quasigeodesic flow. We prove that in the
universal cover, the lifted leaves of $\tilde F$ extend continuously
to the sphere at infinity, giving a continuous
parametrization of their limit sets. This applies for
instance to every Reebless finite depth foliation
in hyperbolic 3-manifolds, which exist whenever
the second Betti number is non zero. It also applies
to large classes of foliations with all leaves dense
and to infinitely many foliations with one sided
branching. One important tool is a careful analysis
of one dimensional singular foliations induced
in the leaves of $F$ or $\tilde F$ by the stable/unstable
singular foliations of the flow $A$.