The fish has no bones
type: Item
instance of: scholarly article
title: The fish has no bones
zbMATH Open document ID: 0971.37001
DOI: 10.1016/S0246-0203(00)00132-1
author: Thierry Bousch
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 2001-03-27T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77669
review text: Let \(\mathcal M\) be the set of Borel probabilities on the unit circle which are invariant with respect to the mapping \(x\mapsto 2x\). It is proved that for each function \(\rho_{\omega}(t):=\cos (2\pi (t-\omega))\), there is exactly one element \(\mu\in {\mathcal M}\) which maximizes \(\smallint\rho_{\omega} d\mu\), and that the support of this measure is contained in a semicircle. In particular, the image of the map \(\mu\mapsto\int\exp\{ 2i\pi t\} d\mu\) is compact and a convex set of the complex plane which does not contain any line segments on its boundary. It is also proved that the maximizing measure is periodic for every \(\omega\) except on a set which has measure zero and Hausdorff dimension zero.
Mathematics Subject Classification ID: 37A05
zbMATH DE Number: 1525675
zbMATH Keywords: Sturm measure
zbMATH Keywords: Sturm condition
zbMATH Keywords: periodic orbits
zbMATH Keywords: invariant measure
reviewed by: Rimas Norvaiša
MaRDI profile type: Publication
OpenAlex ID: W2081629173
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description: scientific article; zbMATH DE number 1525675