An orbit is called `generic' for a flow on a non-compact space, if it satisfies the conclusion of the ratio ergodic theorem for all continuous test functions of compact support and non-zero integral. Furstenberg, Dani & Smillie, and Burger describe the generic orbits for horocycle flows on most hyperbolic surfaces of finite genus. I will give the first characterization of such orbits in an infinite genus setting: abelian covers of compact surfaces. For such surfaces a horocycle is generic iff its associated geodesic has an asymptotic cycle, and this asymptotic cycle is not on the boundary of the set of all possible asymptotic cycles. (Joint work with B. Schapira) |