Limits of geodesic push-forwards of horocycle invariant measures

Abstract

AbstractWe prove several general conditional convergence results on ergodic averages for horocycle and geodesic subgroups of any continuous $\operatorname {SL}(2, \mathbb {R})$ -action on a locally compact space. These results are motivated by theorems of Eskin, Mirzakhani and Mohammadi on the $\operatorname {SL}(2, \mathbb {R})$ -action on the moduli space of Abelian differentials. By our argument we can derive from these theorems an improved version of the ‘weak convergence’ of push-forwards of horocycle measures under the geodesic flow and a short proof of weaker versions of theorems of Chaika and Eskin on Birkhoff genericity and Oseledets regularity in almost all directions for the Teichmüller geodesic flow.