Ergodicity of the absolute period foliation
type: Item
instance of: scholarly article
title: Ergodicity of the absolute period foliation
zbMATH Open document ID: 1398.32016
author: Ursula Hamenstädt
published in: Israel Journal of Mathematics
publication date: 2018-07-03T00:00:00Z
review text: One can define translation surfaces as surfaces obtained by gluing parallel sides of a polygon: namely if \(e_1, \ldots, e_n\) are nonzero vectors in \(\mathbb C\) and \(P\) is a polygon with sides \(e_1, \ldots, e_n\) each occuring exactly twice (in no particular order) then identifying the two sides supported by \(e_i\) for all \(i\) gives a closed surface \(S\). It has the additional structure of a flat metric, possibly with conical points with angles in \(2\pi\mathbb N\), at the vertices of \(P\); to simplify let us assume that all vertices give rise to cone points. Moreover singular chains coming from \(e_1, \ldots, e_n\) give a basis for the relative homology \(H_1(S, V)\) where \(V\) is the set of vertices of \(P\) giving rise to cone points. There is a natural decomposition of the moduli space of such surfaces into strata corresponding to the orders of cone points. Generically there are only cone points of angle \(4\pi\) (the minimal possible); the stratum corresponding to such surfaces is called the principal stratum. It is also possible to define a translation surface as a pair \((X, \omega)\) where \(X\) is a Riemann surface and \(\omega\) a holomorphic differential (also called an ``abelian differential''). The angle of the cone points then corresponds to the order of zeroes of \(\omega\). Integrating \(\omega\) against the relative cycles defined above gives local coordinates for the moduli space of such surfaces. The absolute homology \(H_1(S)\) injects into \(H_1(S, V)\). Integrals against absolute classes are called absolute periods, and the absolute period foliation is given by level sets of absolute periods; that is a leaf is a maximal subset where all such periods are constant. Such leaves are contained in strata of the decomposition described above. \textit{C. T. McMullen} [Duke Math. J. 163, No. 12, 2271--2323 (2014; Zbl 1371.30037)] proved that if \(S\) has genus 2 or 3 this foliation is ergodic on the principal stratum, meaning that every Bore…
reviewed by: Jean Raimbault
Mathematics Subject Classification ID: 32G15
Mathematics Subject Classification ID: 37D50
zbMATH DE Number: 6898289
zbMATH Keywords: moduli space
zbMATH Keywords: translation surface
zbMATH Keywords: period foliation
MaRDI profile type: Publication
full work available at URL: s11856-018-1674-4
OpenAlex ID: W2799795844
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Wikidata QID: Q129915925
DOI: 10.1007/S11856-018-1674-4
description: scientific article; zbMATH DE number 6898289