Closed trajectories for quadratic differentials with an application to billiards
type: Item
instance of: scholarly article
title: Closed trajectories for quadratic differentials with an application to billiards
zbMATH Open document ID: 0616.30044
DOI: 10.1215/S0012-7094-86-05319-6
author: Howard Masur
published in: Duke Mathematical Journal
publication date: 1986-01-01T00:00:00Z
review text: It is proved that for any given holomorphic quadratic differential q on a compact Riemann surface of genus \(g\geq 2\), there exists a dense set of \(\theta\) for which \(e^{i\theta}q\) has a closed regular vertical trajectory. This result is relative to a problem of the dynamical system of a polygonal table with angles, rational multiples of \(\pi\), the so called rational billiards. In a paper of \textit{C. Boldrighini, C. Keane} and \textit{F. Marchetti} [Ann. Prob. 6, 532-540 (1978; Zbl 0377.28014)] the question was raised whether every rational billiard has a periodic orbit. As an application of the above result of the present paper, it is proved that for any rational billiard table there is a dense set of directions each with a periodic orbit.
Mathematics Subject Classification ID: 30F30
Mathematics Subject Classification ID: 30F99
Mathematics Subject Classification ID: 32G15
zbMATH DE Number: 3998189
zbMATH Keywords: quadratic differential
zbMATH Keywords: dynamical system
zbMATH Keywords: rational billiards
MaRDI profile type: Publication
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full work available at URL: s0012-7094-86-05319-6
OpenAlex ID: W1496711271
description: scientific article; zbMATH DE number 3998189