{"title":"Rotation number of contracted rotations","authors":"M. Laurent, A. Nogueira","doi":"10.3934/JMD.2018007","DOIUrl":null,"url":null,"abstract":"Let \\begin{document} $0 . We consider the one-parameter family of circle \\begin{document} $\\lambda$ \\end{document} -affine contractions \\begin{document} $f_\\delta:x \\in [0,1) \\mapsto \\lambda x + \\delta \\; {\\rm mod}\\,1 $ \\end{document} , where \\begin{document} $0 \\le \\delta . Let \\begin{document} $\\rho$ \\end{document} be the rotation number of the map \\begin{document} $f_\\delta$ \\end{document} . We will give some numerical relations between the values of \\begin{document} $\\lambda,\\delta$ \\end{document} and \\begin{document} $\\rho$ \\end{document} , essentially using Hecke-Mahler series and a tree structure. When both parameters \\begin{document} $\\lambda$ \\end{document} and \\begin{document} $\\delta$ \\end{document} are algebraic numbers, we show that \\begin{document} $\\rho$ \\end{document} is a rational number. Moreover, in the case \\begin{document} $\\lambda$ \\end{document} and \\begin{document} $\\delta$ \\end{document} are rational, we give an explicit upper bound for the height of \\begin{document} $\\rho$ \\end{document} under some assumptions on \\begin{document} $\\lambda$ \\end{document} .","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":"12 1","pages":"175-191"},"PeriodicalIF":0.7000,"publicationDate":"2018-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"13","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Modern Dynamics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3934/JMD.2018007","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 13
Abstract
Let \begin{document} $0 . We consider the one-parameter family of circle \begin{document} $\lambda$ \end{document} -affine contractions \begin{document} $f_\delta:x \in [0,1) \mapsto \lambda x + \delta \; {\rm mod}\,1 $ \end{document} , where \begin{document} $0 \le \delta . Let \begin{document} $\rho$ \end{document} be the rotation number of the map \begin{document} $f_\delta$ \end{document} . We will give some numerical relations between the values of \begin{document} $\lambda,\delta$ \end{document} and \begin{document} $\rho$ \end{document} , essentially using Hecke-Mahler series and a tree structure. When both parameters \begin{document} $\lambda$ \end{document} and \begin{document} $\delta$ \end{document} are algebraic numbers, we show that \begin{document} $\rho$ \end{document} is a rational number. Moreover, in the case \begin{document} $\lambda$ \end{document} and \begin{document} $\delta$ \end{document} are rational, we give an explicit upper bound for the height of \begin{document} $\rho$ \end{document} under some assumptions on \begin{document} $\lambda$ \end{document} .
期刊介绍:
The Journal of Modern Dynamics (JMD) is dedicated to publishing research articles in active and promising areas in the theory of dynamical systems with particular emphasis on the mutual interaction between dynamics and other major areas of mathematical research, including:
Number theory
Symplectic geometry
Differential geometry
Rigidity
Quantum chaos
Teichmüller theory
Geometric group theory
Harmonic analysis on manifolds.
The journal is published by the American Institute of Mathematical Sciences (AIMS) with the support of the Anatole Katok Center for Dynamical Systems and Geometry at the Pennsylvania State University.