Dynamical properties of 2-torus parabolic maps
type: Item
instance of: scholarly article
title: Dynamical properties of 2-torus parabolic maps
zbMATH Open document ID: 1185.37039
author: Xinchu Fu
author: Fangyue Chen
author: X. H. Zhao
published in: Nonlinear Dynamics
publication date: 2009-09-22T00:00:00Z
review text: The authors consider a parabolic map of the torus \(f:[0,1)^2 \rightarrow [0,1)^2\) where \(f(x,y)=(x',y')\) is defined by \[ \begin{cases} x'=ax+by \pmod1\\ y'=cx+dy \pmod1. \end{cases} \] Let \(X^+=\bigcap_{n=0}^\infty f^n(x)\). First it is shown that \(X^+\) is invariant and almost closed and that the Lebesgue measure restricted to \(X^+\) is invariant. For some examples, the Lebesgue measure of \(X^+\) is computed and estimated [see also \textit{P. Ashwin, X.C. Fu, T. Nishikawa}, and \textit{K. Zyczkowski}, Nonlinearity 13, No. 3, 819--835 (2000; Zbl 0952.37007)]. When \(f\) is a non-horocyclic map and \(f^{-1}\) exists, it is shown that \(f^{-1}\) is not parabolic. Furthermore, the conjugation of invertible maps with integral coefficients is studied. The authors show that this class of maps can be reduced to a family of one-parameter rigid rotations.
Mathematics Subject Classification ID: 37C15
Mathematics Subject Classification ID: 37D50
zbMATH DE Number: 5605879
zbMATH Keywords: parabolic map
zbMATH Keywords: maximal invariant set
zbMATH Keywords: invariant measure
zbMATH Keywords: conjugation invariant
reviewed by: Steve Pederson
MaRDI profile type: Publication
full work available at URL: s11071-006-9179-9
OpenAlex ID: W2029096025
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DOI: 10.1007/S11071-006-9179-9
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description: scientific article; zbMATH DE number 5605879