Limits of geodesic push-forwards of horocycle invariant measures

TL;DR

This paper establishes general convergence results for ergodic averages of horocycle and geodesic actions in SL(2,R) dynamics, improving understanding of measure push-forwards and their convergence properties.

Contribution

It provides new conditional convergence theorems for ergodic averages in SL(2,R) actions, extending and refining previous results on measure dynamics in moduli spaces.

Findings

Improved weak convergence of horocycle measure push-forwards under geodesic flow.

Simplified proofs of Birkhoff genericity and Oseledets regularity for Teichmüller flow.

General conditional convergence results applicable to various SL(2,R) actions.

Abstract

We prove several general conditional convergence results on ergodic averages for horocycle and geodesic subgroups of any continuous action of the Lie group SL(2, R) on a locally compact space. These results are motivated by theorems of Eskin, Mirzakhani and Mohammadi on the SL(2,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 Teichmueller geodesic flow.