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

Yair N. Minsky; Barak Weiss

Home/Archive/arXiv

arXiv · 1102.4719

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

Yair N. Minsky·Barak Weiss

View original source ↗Read source document ↗Export metadata ↓

Abstract

A translation surface on (S, Σ) gives rise to two transverse measured foliations \FF, \GG on S with singularities in Σ, and by integration, to a pair of cohomology classes [\FF], \, [\GG] \in H^1(S, Σ; \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 σ\in \mathcal{S}_d, the space \R^d_+ parametrizes the interval exchanges on d intervals with permutation σ. We describe lines \ell in \R^d_+ such that almost every point in \ell is uniquely ergodic. We also show that for σ(i) = d+1-i, for almost every s>0, the interval exchange transformation corresponding to σand (s, s^2, \ldots, s^d) is uniquely ergodic. As another application we show that when k=|Σ| \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, Σ) without horizontal saddle connections. Here B \subset \SL(2,\R) is the subgroup of upper triangular matrices.

Explore related subjects

math.DS

Keep this discovery

BibTeX RIS

Yair N. Minsky, Barak Weiss. 2011-02-23. Cohomology classes represented by measured foliations, and Mahler's question for interval exchanges. https://arxiv.org/abs/1102.4719

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Admissible Fourier Lengths, KAM Reducibility, and Spectral Applications

We develop a perturbative KAM reducibility theory for one-frequency $\mathrm{SL}(2,\mathbb{R})$ cocycles based on an admissible Fourier length $\ell$. The regularity relevant to the iteration is measured by positive adapted Fourier width rather than ordinary smoothness in the Euclidean length $|n|$. The same length governs Fourier decay, truncation and resonance scales, and the arithmetic condition controlling the small divisors. This framework contains the classical analytic and Gevrey settings, while non-monotone choices of $\ell$ allow classical nowhere differentiable Weierstrass-type perturbations and continuous perturbations outside every positive H\"older class. As spectral applications, we obtain purely absolutely continuous spectrum for every phase and $1/2$-H\"older continuity of the integrated density of states for the associated quasiperiodic Schr\"odinger operators. The Aubry dual has pure point spectrum for Lebesgue almost every dual phase, with eigenfunctions exponentially localized in the metric induced by $\ell$. We also construct nowhere differentiable quasiperiodic potentials with purely absolutely continuous Cantor spectrum.

math.DS↗

arXiv2026

Dynamics inside the attracting basins of some skew products

Polynomial skew products in $\mathbb{C}^2$ are maps of the form $F(z,w)=(P(z),Q(z,w))$, where $P$ and $Q$ are polynomials. Their local dynamics have been widely investigated. In this paper, we study the global dynamics inside Fatou components of some skew products. We consider all the inverse images in a Fatou component of a given point and use the Kobayashi metric to measure the distance between points. In the cases we consider, there are always arbitrarily large Kobayashi balls in the complement of these inverse sets.

math.DS↗

arXiv2026

Ergodicity of dynamical systems without uniqueness of orbits

Recently, there has been considerable interest in the study of non-deterministic dynamical systems. To analyze the chaotic behavior of such systems from a measure-theoretic viewpoint, it is desirable to consider ergodicity. However, the classical definition of ergodicity involves invariant sets, whose definition is not unique for non-deterministic dynamical systems. Thus, we are led to the question of which invariance yields an interesting definition of ergodicity. Here, we propose a definition based on the strong backward invariance and show that analogs of classical results hold. We also consider implications of the Birkhoff ergodic theorem for systems without uniqueness of orbits.

math.DS↗