(Joint work with James Farre and Or Landesberg)

Horospherical flow is an important phenomenon in dynamics. It can be very rigid: since the work of Hedlund in the 30's it is known, for example, that horocycles in closed hyperbolic surfaces are always dense and equidistributed. In infinite-volume manifolds there is much more flexibility. The case of Z-covers of a closed hyperbolic manifold M has a particularly nice blend of flexibility and rigidity. There is an interesting connection between closures of horospherical orbits and optimal 1-Lipschitz maps from M to a circle. Such maps achieve their maximal stretch on a geodesic lamination, and the dynamics and geometry of this lamination influences the behavior of the orbit closures. We can give a complete description of horocycle orbit closures in the case of Z-covers of a closed hyperbolic surface. In particular we were surprised to find that all proper orbit closures are fractal in some sense, and yet have integer Hausdorff dimension.