Joining measures for horocycle flows on abelian covers

Wenyu Pan

doi:10.3934/jmd.2018003

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

This volume Previous Article Rational ergodicity of step function skew products Next Article Symbolic dynamics for non-uniformly hyperbolic diffeomorphisms of compact smooth manifolds

Joining measures for horocycle flows on abelian covers

Wenyu Pan

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

Received: January 31, 2017

Revised: November 09, 2017

Published: March 2018

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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.

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Mathematics Subject Classification: Primary: 37A17, 37D40; Secondary: 37D35.

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