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

TL;DR

This paper studies invariant measures under SL(2,R) actions on vector bundles, showing they concentrate on the top Lyapunov subspace, with applications to Lyapunov exponents' continuity and ergodicity of foliated flows.

Contribution

It proves invariance of measures under upper triangular groups implies support on the top Lyapunov subspace, extending understanding of cocycles and invariant measures.

Findings

Invariant measures supported on top Lyapunov subspace

Lyapunov exponents depend continuously on affine measures

Foliated horocycle flow is uniquely ergodic under certain conditions

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.