Projective cocycles over SL(2,R) actions: measures invariant under the upper triangular group

2017/09/08 by Christian Bonatti, Bonatti, Christian, Alex Eskin +3 · 1 citation
Mathematics · #37A05 #37C40 #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals

paper · doi:10.48550/arxiv.1709.02521

openalex publication_date 2017/09/08 · openalex created_date 2022/09/01 · openalex updated_date 2026/07/28

Abstract

We consider the action of SL(2,ℝ) on a vector bundle H preserving an ergodic probability measure ν on the base X. Under an irreducibility assumption on this action, we prove that if ν is any lift of ν to a probability measure on the projectivized bunde ℙ(H) that is invariant under the upper triangular subgroup, then ν is supported in the projectivization ℙ(E₁) of the top Lyapunov subspace of the positive diagonal semigroup. We derive two applications. First, the Lyapunov exponents for the Kontsevich-Zorich cocycle depend continuously on affine measures, answering a question in [MMY]. Second, if ℙ(V) is an irreducible, flat projective bundle over a compact hyperbolic surface Σ, with hyperbolic foliation F tangent to the flat connection, then the foliated horocycle flow on T¹F is uniquely ergodic if the top Lyapunov exponent of the foliated geodesic flow is simple. This generalizes results in [BG] to arbitrary dimension.

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