A pair of disjoint invariant path-connected sets on the boundary of a hyperbolic 3-manifold canonically yields a universal circle with invariant laminations.
Mosher,Laminations and flows transverse to finite depth foliations, Part I: branched surfaces and dynamics, preprint (1996), available athttp://newark.rutgers.edu:80/~mosher
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A pair of disjoint invariant path-connected sets on the boundary of a hyperbolic 3-manifold canonically yields a universal circle with invariant laminations.