On piecewise class \(PC^r\) homeomorphisms of the circle which are piecewise \(C^ r (r\geq 1)\) conjugate to irrational rotations
type: Item
instance of: scholarly article
title: On piecewise class \(PC^r\) homeomorphisms of the circle which are piecewise \(C^ r (r\geq 1)\) conjugate to irrational rotations
zbMATH Open document ID: 1147.37024
author: Abdelhamid Adouani
author: Habib Marzougui
publication date: 2008-07-04T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10333
review text: It is a well known fact that any orientation-preserving diffeomorphism of the circle \(S^1\) of class \(C^r\), \(r\geq 2,\) having an irrational rotation number \(\rho\) is topologically conjugate to the rotation of angle \(\rho\). The same applies for a more general class of homeomorphisms, the class \(\mathcal{P}\) composed by orientation-preserving homeomorphisms \(f\) which are differentiable except for a countable set of ``break points'' holding some appropriate conditions of boundedness respect their right and left derivatives. Inside \(\mathcal{P}\) the authors consider the so-called class \(\mathcal{P}^r(S^1), r\geq 1,\) of homeomorphisms \(f\) such that \(f\) is \(C^{[r]}\) (\([\cdot]\) is meant the integer part of \(r\)) except for a finite number of points (singular points) for which it is assumed the existence of right and left derivatives up to the order \(n,\) and such that \(D^{[r]}f\) satisfies a Hölder condition of order \(r-[r]\) in any interval \(I\subset S^1\) where \(f\) is \(C^{[r]}.\) In the main result of the present paper the authors characterize the elements of \(\mathcal{P}^r(S^1)\) having an irrational rotation number which are topologically conjugate to a \(C^r\)-diffeomorphism via a topological condition, named \(D_r\) (too much involved to be described here), related with the differentiability of certain iterates of \(f\) in the singular points. Moreover, the conjugacy homeomorphism is proved to be inside \(\mathcal{P}^r(S^1)\) and to be a piecewise polynomial. Additionally, if the rotation number \(\rho\) of \(f\) is Diophantine of order \(\tau \geq 0\) , that is \(| \rho -\frac{p}{q}| >\frac{C}{q^{2+\tau}}\) for any rational number \(\frac{p}{q}\) and some positive constant \(C,\) then \(f\) is topologically conjugate to the rotation \(R_{\rho}\) by a homeomorphism which is piecewise of class \(C^{r-1-\tau\varepsilon}\) for all \(\varepsilon >0\). This result can be viewed as a generalization of another stated in \textit{A. A. Dzhali…
Mathematics Subject Classification ID: 37E10
Mathematics Subject Classification ID: 37C15
Mathematics Subject Classification ID: 37C40
zbMATH DE Number: 5298320
zbMATH Keywords: circle maps
zbMATH Keywords: piecewise class \(PC^r\) homeomorphism
zbMATH Keywords: Hölder condition
zbMATH Keywords: rotation number
zbMATH Keywords: conjugacy
zbMATH Keywords: break point
zbMATH Keywords: singular point
zbMATH Keywords: jump
zbMATH Keywords: invariant measure
zbMATH Keywords: equivalent measure
zbMATH Keywords: singular measure
reviewed by: Antonio Linero Bas
MaRDI profile type: Publication
OpenAlex ID: W2460613079
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DOI: 10.5802/AIF.2368
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 5298320