  Let F be a foliation in a closed 3-manifold with negatively curved
fundamental group and suppose that F is almost transverse to a quasigeodesic
pseudo-Anosov flow. We show that the leaves of the foliation in the universal
cover extend continuously to the sphere at infinity, hence the limit sets are
continuous images of the circle. One important corollary is that if F is a
Reebless finite depth foliation in a hyperbolic manifold, then it has the
continuous extension property. Such finite depth foliations exist whenever the
second Betti number is non zero. The result also applies to other classes of
foliations, including a large class of foliations where all leaves are dense
and infinitely many examples with one sided branching. One key tool is a
detailed understanding of asymptotic properties of almost pseudo-Anosov
singular 1-dimensional foliations in the leaves of F lifted to the universal
cover.
