# American Institute of Mathematical Sciences

2018, 12: 17-54. doi: 10.3934/jmd.2018003

## Joining measures for horocycle flows on abelian covers

 Department of Mathematics, Yale University, New Haven, CT 06511, USA

Received  January 31, 2017 Revised  November 09, 2017 Published  March 2018

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.

Citation: Wenyu Pan. Joining measures for horocycle flows on abelian covers. Journal of Modern Dynamics, 2018, 12: 17-54. doi: 10.3934/jmd.2018003
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