The Generic Points for the Horocycle Flow on a Class of Hyperbolic Surfaces with Infinite Genus

A point is called generic for a flow preserving an infinite ergodic invariant Radon measure, if its orbit satisfies the conclusion of the ratio ergodic theorem for every pair of continuous functions with compact support and nonzero integrals. The generic points for horocycle flows on hyperbolic surfaces of finite genus are understood, but there are no results in infinite genus. We give such a result by characterizing the generic points for Z(d)-covers.