1. Projective cocycles over SL(2,R) actions: measures invariant under the upper triangular group
- Author
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Amie Wilkinson, Alex Eskin, Christian Bonatti, Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), Department of Mathematics [Chicago], University of Chicago, and National Science Foundation (NSF)NSF - Directorate for Mathematical & Physical Sciences (MPS)1402852
- Subjects
Projectivization ,Pure mathematics ,Mathematics::Dynamical Systems ,General Mathematics ,Triangular matrix ,Vector bundle ,nonuniform hyperbolicity ,Dynamical Systems (math.DS) ,quadratic-differentials ,Lyapunov exponent ,abelian differentials ,surfaces ,simplicity ,teichmuller curves ,01 natural sciences ,spectrum ,projective cocycles ,symbols.namesake ,0103 physical sciences ,FOS: Mathematics ,Ergodic theory ,[MATH]Mathematics [math] ,Mathematics - Dynamical Systems ,0101 mathematics ,Invariant (mathematics) ,Mathematics ,Probability measure ,010102 general mathematics ,Lyapunov exponents ,criterion ,parabolic group actions ,Horocycle ,symbols ,moduli spaces ,37C40, 37A05 ,Mathematics::Differential Geometry ,010307 mathematical physics ,Invariant measures ,zero lyapunov exponents - Abstract
We consider the action of $SL(2,\mathbb{R})$ on a vector bundle $\mathbf{H}$ preserving an ergodic probability measure $\nu$ on the base $X$. Under an irreducibility assumption on this action, we prove that if $\hat\nu$ is any lift of $\nu$ to a probability measure on the projectivized bunde $\mathbb{P}(\mathbf{H})$ that is invariant under the upper triangular subgroup, then $\hat \nu$ is supported in the projectivization $\mathbb{P}(\mathbf{E}_1)$ 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 $\mathbb{P}(\mathbf{V})$ is an irreducible, flat projective bundle over a compact hyperbolic surface $\Sigma$, with hyperbolic foliation $\mathcal{F}$ tangent to the flat connection, then the foliated horocycle flow on $T^1\mathcal{F}$ is uniquely ergodic if the top Lyapunov exponent of the foliated geodesic flow is simple. This generalizes results in [BG] to arbitrary dimension., Comment: Minor corrections. 24 pages, 1 figure
- Published
- 2020