
Mathematics 2011
Cohomology classes represented by measured foliations, and Mahler's question for interval exchangesAbstract: 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+1i, 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 welldefined. We prove that there is a welldefined action of the group B \ltimes \R^{k1} 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.
