Rotation number, invariant measures and ratio set of PL homeomorphisms of the circle.
type: Item
instance of: scholarly article
title: Rotation number, invariant measures and ratio set of PL homeomorphisms of the circle.
zbMATH Open document ID: 1079.37033
author: Isabelle Liousse
publication date: 2005-05-25T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116196
review text: The paper deals with piecewise affine circle homeomorphisms without periodic points. This class of maps was studied by M. Herman who found examples of transformations of this class with singular invariant measures for any rotation number. On the other hand, it is not difficult to find examples in which points of discontinuity of the derivative meet other such points and the jumps of the derivative cancel out. The present paper shows that, at least under some circumstances, this is the only way to get an absolutely continuous invariant measure, whose density then turns out to be piecewise analytic. The key condition is that that in order for the conjugacy to be piecewise \(C^1\), the number of points of non-differentiability must remain uniformly bounded for all iterates of the map. It would be interesting to know whether any of these results remain true for mappings which are piecewise smooth, but not affine, and for which the product of all jumps of the derivative is 1.
Mathematics Subject Classification ID: 37E10
Mathematics Subject Classification ID: 37C15
Mathematics Subject Classification ID: 37E45
Mathematics Subject Classification ID: 37C40
zbMATH DE Number: 2171514
zbMATH Keywords: denjoy theory
zbMATH Keywords: jump singularity
zbMATH Keywords: singular conjugacy
reviewed by: Grzegorz Świątek
MaRDI profile type: Publication
OpenAlex ID: W2470590826
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DOI: 10.5802/AIF.2103
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 2171514