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

Abstract

A translation surface on (S;�) gives rise to two transverse measured foliations F;G on S with singularities in �, and by integration, to a pair of cohomology classes (F);(G) 2 H 1 (S;�;R). Given a measured foliation F, we characterize the set of cohomology classes b for which there is a measured foliation G as above with b = (G). This extends previous results of Thurston (Th) and Sullivan (Su). We apply this to two problems: unique ergodicity of interval ex- changes and flows on the moduli space of translation surfaces. For a fixed permutation � 2 Sd, the space R d parametrizes the interval ex- changes on d intervals with permutation �. We describe linesin R d such that almost every point inis uniquely ergodic. We also show that for �(i) = d+1 i, for almost every s > 0, the interval exchange trans- formation corresponding toand (s;s 2 ;:::;s d ) is uniquely ergodic. As another application we show that when k = |�| � 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 n R k 1 on the set of translation surfaces of type (S;�) without horizontal saddle connections. Here BSL(2;R) is the subgroup of upper triangular matrices.