Invariant measures and stiffness for non-Abelian groups of toral automorphisms

Abstract

Let Γ be a non-elementary subgroup of SL 2 ( Z ) . If μ is a probability measure on T 2 which is Γ-invariant, then μ is a convex combination of the Haar measure and an atomic probability measure supported by rational points. The same conclusion holds under the weaker assumption that μ is ν-stationary, i.e. μ = ν ∗ μ , where ν is a finitely supported, probability measure on Γ whose support supp ν generates Γ. The approach works more generally for Γ < SL d ( Z ) . To cite this article: J. Bourgain et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007).