The generic points for the horocycle flow on a class of hyperbolic surfaces with infinite genus

TL;DR

This paper characterizes generic points for horocycle flows on hyperbolic surfaces with infinite genus, extending understanding from finite to infinite genus cases using ratio ergodic theorem concepts.

Contribution

It provides the first characterization of generic points for horocycle flows on infinite genus hyperbolic surfaces, specifically for -covers.

Findings

Characterization of generic points for -covers

Extension of ratio ergodic theorem to infinite genus surfaces

New insights into ergodic properties of flows on complex surfaces

Abstract

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 non-zero 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.