## Abstract

A translation structure on (S, Σ) gives rise to two transverse measured foli- ations F, G on S with singularities in Σ, and by integration, to a pair of relative cohomology classes [F]; [G] ε H ^{1} (S, Σ,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 [19] and Sullivan [18]. We apply this to two problems: unique ergodicity of interval exchanges and flows on the moduli space of translation surfaces. For a fixed permutation σ ε Sd, the space ℝ _{+}^{d} parametrizes the interval exchanges on d intervals with permutation ℓ.We describe lines ℓ in ℝ _{+}^{d} such that almost every point in ℓ is uniquely ergodic.We also show that for σ(i) = d+1+i, for almost every s > 0, the interval exchange transformation corresponding to σ and (s, s2, : : :, sd) is uniquely ergodic. As another application we show that when k = |Σ| ≥ 2; the operation of "moving the singularities horizontally" is globally well-defined.We prove that there is a well-defined action of the groupB×ℝk-1 on the set of translation surfaces of type (S, Σ) without horizontal saddle connections. Here B ⊂ SL(2, ℝ) is the subgroup of upper triangular matrices.

Original language | English |
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Pages (from-to) | 245-284 |

Number of pages | 40 |

Journal | Annales Scientifiques de l'Ecole Normale Superieure |

Volume | 47 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jan 2014 |

## ASJC Scopus subject areas

- Mathematics (all)