The horocycle flow on hyperbolic surfaces has attracted considerable attention in the last century. In the '30s, Hedlund proved that all horocycle orbits are dense in closed hyperbolic surfaces, and the classification problem for horocycle orbit closures has been solved for geometrically finite surfaces. We are interested in the topology and dynamics of horocycle orbits in the geometrically infinite setting, where our understanding is much more limited.
In this talk, I will discuss joint work with Or Landesberg and Yair Minsky: we give the first complete classification of orbit closures for a class of Z-covers of closed surfaces. Our analysis is rooted in a seemingly unrelated geometric optimization problem: finding a best Lipschitz map to the circle. We then relate the topology of horocycle orbit closures with the dynamics of the minimizing lamination of maximal stretch, as studied by Guéritaud-Kassel and Daskalopoulos-Uhlenbeck.