Cohomology classes represented by measured foliations, and Mahler's question for interval exchanges

TL;DR

This paper characterizes cohomology classes represented by measured foliations on translation surfaces, extending Thurston and Sullivan's work, and applies these results to problems in ergodic theory and moduli space dynamics, including unique ergodicity of interval exchanges.

Contribution

It extends the characterization of cohomology classes of measured foliations and applies this to analyze unique ergodicity and the dynamics of translation surfaces.

Findings

Almost every point on certain lines in parameter space is uniquely ergodic.

Interval exchanges with specific permutations are uniquely ergodic for almost all parameters.

The operation of moving singularities horizontally is globally well-defined and induces a group action.

Abstract

A translation surface on (S, \Sigma) gives rise to two transverse measured foliations \FF, \GG on S with singularities in \Sigma, and by integration, to a pair of cohomology classes [\FF], \, [\GG] \in H^1(S, \Sigma; \R). Given a measured foliation \FF, we characterize the set of cohomology classes \B for which there is a measured foliation \GG as above with \B = [\GG]. This extends previous results of Thurston and Sullivan. We apply this to two problems: unique ergodicity of interval exchanges and flows on the moduli space of translation surfaces. For a fixed permutation \sigma \in \mathcal{S}_d, the space \R^d_+ parametrizes the interval exchanges on d intervals with permutation \sigma. We describe lines \ell in \R^d_+ such that almost every point in \ell is uniquely ergodic. We also show that for \sigma(i) = d+1-i, for almost every s>0, the interval exchange transformation corresponding to \sigma and (s, s^2, \ldots, s^d) is uniquely ergodic. As another application we show that when k=|\Sigma| \geq 2, the operation of `moving the singularities horizontally' is globally well-defined. We prove that there is a well-defined action of the group B \ltimes \R^{k-1} on the set of translation surfaces of type (S, \Sigma) without horizontal saddle connections. Here B \subset \SL(2,\R) is the subgroup of upper triangular matrices.