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Register a new userInvariant and stationary measures for the SL(2,R) action on Moduli space
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We prove some ergodic-theoretic rigidity properties of the action of SL(2,R) on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of SL(2,R) is supported on an invariant affine submanifold. The main theorems are inspired by the results of several authors on unipotent flows on homogeneous spaces, and in particular by Ratners seminal work.
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We prove results about orbit closures and equidistribution for the SL(2,R) action on the moduli space of compact Riemann surfaces, which are analogous to the theory of unipotent flows. The proofs of the main theorems rely on the measure classification theorem of [EMi2] and a certain isolation property of closed SL(2,R) invariant manifolds developed in this paper.
We consider the action of $SL(2,mathbb{R})$ on a vector bundle $mathbf{H}$ preserving an ergodic probability measure $ u$ on the base $X$. Under an irreducibility assumption on this action, we prove that if $hat u$ is any lift of $ u$ to a probability measure on the projectivized bunde $mathbb{P}(mathbf{H})$ that is invariant under the upper triangular subgroup, then $hat u$ 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 $T^1mathcal{F}$ is uniquely ergodic if the top Lyapunov exponent of the foliated geodesic flow is simple. This generalizes results in [BG] to arbitrary dimension.
Suppose N is an affine SL(2,R)-invariant submanfold of the moduli space of pairs (M,w) where M is a curve, and w is a holomorphic 1-form on M. We show that the Forni bundle of N (i.e. the maximal SL(2,R)-invariant isometric subbundle of the Hodge bundle of N) is always flat and is always orthogonal to the tangent space of N. As a corollary, it follows that the Hodge bundle of N is semisimple.
We compute the asymptotics, as R tends to infinity, of the number of closed geodesics in Moduli space of length at most R, or equivalently the number of pseudo-Anosov elements of the mapping class group of translation length at most R.
We study the asymptotic behavior of the Lyapunov exponent in a meromorphic family of random products of matrices in SL(2, C), as the parameter converges to a pole. We show that the blow-up of the Lyapunov exponent is governed by a quantity which can be interpreted as the non-Archimedean Lyapunov exponent of the family. We also describe the limit of the corresponding family of stationary measures on P 1 (C).
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