Limits of geodesic push-forwards of horocycle invariant measures

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.

22 pages. A few misprints have been corrected, and one reference has been updated

Cited by in corpus (4)

• Shear-shape cocycles for measured laminations and ergodic theory of the earthquake flow
• Continuity of the orthogeodesic foliation and ergodic theory of the earthquake flow
• The distribution of critical graphs of Jenkins-Strebel differentials
• On Convergence of Random Walks on Moduli Space