Abstract: The geodesic flow on the unit tangent bundle of a complete hyperbolic surface is a prototypical example of an Anosov flow. While invariant sets and measures for the geodesic flow can be quite complicated, there is a closely related flow, the horocycle flow, where the analogous objects tend to be better behaved. The orbits of the horocycle flow are the strong stable manifolds for the geodesic flow.
In this talk, I will discuss the behavior of horocycle orbits in Z-covers of closed hyperbolic surfaces and their relationship to an exceptional set for the geodesic flow, the distance minimizing geodesic lamination. Both objects are closely related to a certain geometric optimization problem: finding a best Lipschitz map to the circle in a given homotopy class (defining the Z-cover). This is joint work with Or Landesberg and Yair Minsky.
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