Joining measures for horocycle flows on abelian covers

TL;DR

This paper classifies joinings of horocycle flows on certain infinite-genus hyperbolic surfaces, extending Ratner's and Mohammadi-Oh's results to new classes of infinite-area surfaces.

Contribution

It provides the first joining classification for horocycle flows on infinite-genus hyperbolic surfaces, specifically $ ext{Z}$ or $ ext{Z}^2$-covers.

Findings

Classification of joinings for $ ext{Z}$ or $ ext{Z}^2$-covers of compact hyperbolic surfaces.

Extension of Ratner's and Mohammadi-Oh's theorems to infinite-genus surfaces.

Discussion of applications of the joining classification.

Abstract

A celebrated result of Ratner from the eighties says that two horocycle flows on hyperbolic surfaces of finite area are either the same up to algebraic change of coordinates, or they have no non-trivial joinings. Recently, Mohammadi and Oh extended Ratner's theorem to horocycle flows on hyperbolic surfaces of infinite area but finite genus. In this paper, we present the first joining classification result of a horocycle flow on a hyperbolic surface of infinite genus: a $\mathbb{Z}$ or $\mathbb{Z}^2$-cover of a general compact hyperbolic surface. We also discuss several applications.