Hausdorff dimension of the set of nonergodic foliations of a quadratic differential
type: Item
instance of: scholarly article
title: Hausdorff dimension of the set of nonergodic foliations of a quadratic differential
zbMATH Open document ID: 0780.30032
DOI: 10.1215/S0012-7094-92-06613-0
published in: Duke Mathematical Journal
publication date: 1992-12-14T00:00:00Z
review text: Let \(q=\{q_ z(z)\}\) be a (normalized integrable) quadratic differential on a compact Riemann surface \(X\). That is, \(q_ z(z)dz^ 2\) is invariant under change of local parameters of \(X\) and satisfies \(\int_ X| q_ z(z)dz^ 2|=1\). The metric defined by \(| q^{1/2} dz|\) has singularities at the poles and zeroes of \(q\). Except at these points the metric is Euclidean. A saddle connection of \(q\) means a geodesic segment which joins two singularities of \(q\) and has no singularities in its interior. The vertical trajectories of \(q\) are those curves along which \(q_ z(z)dz^ 2<0\). For any real \(\theta\), \(|\theta|\leq \pi/2\), denote by \(F_ \theta\) the measured foliation whose leaves are the vertical trajectories of the quadratic differential \(e^{2i\theta} q\). A measured foliation \(F\) is called minimal if every closed set which is a union of leaves either is empty or its complement is. Replacing the terms ``closed'' by ``measurable'' and ``empty'' by ``of measure zero'' in the definition of minimality, we have the notion of ergodicity. Let \(\text{NE}(q)\) be the set of \(\theta\) such that \(F_ \theta\) is not ergodic. The author proves that the Hausdorff dimension of \(\text{NE}(q)\) does not exceed \({1\over 2}\). Actually, he proves more: Let \(\text{NUE}(q)\) be the set of \(\theta\) such that \(F_ \theta\) is not uniquely ergodic, that is, \(F_ \theta\) has more than one -- even if we ignore scalar multiplications -- transverse invariant measure. The set \(\text{NUE}(q)\) has Lebesgue measure zero [cf. \textit{S. Kerckhoff}, the author and \textit{J. Smillie}, Ann. Math., II. Ser. 124, No. 2, 293-311 (1986; Zbl 0637.58010)]. The main theorem now reads: The Hausdorff dimension of \(\text{NUE}(q)\) is at most \({1\over 2}\). To prove the main theorem the author first improves an earlier result of his [cf. Ann. Math., II. Ser. 115, No. 1, 169-200 (1982; Zbl 0497.28012)] and shows that if \(f_ t: X\to X_ t\) is the Teichmüller map defined by \(t\) an…
Mathematics Subject Classification ID: 30F30
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 37A99
Mathematics Subject Classification ID: 37C85
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 78545
zbMATH Keywords: quadratic differential
zbMATH Keywords: compact Riemann surface
zbMATH Keywords: Hausdorff dimension
author: Howard Masur
MaRDI profile type: Publication
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full work available at URL: s0012-7094-92-06613-0
OpenAlex ID: W1578008436
description: scientific article; zbMATH DE number 78545