A note on the coding of orbits in certain discontinuous maps
type: Item
instance of: scholarly article
title: A note on the coding of orbits in certain discontinuous maps
zbMATH Open document ID: 1196.37042
published in: Discrete and Continuous Dynamical Systems
publication date: 2010-05-06T00:00:00Z
review text: Let \((X,d)\) be a metric space and \(\{P_1,\dots,P_m\}\) a finite collection of open sets defining a topological partition of \(X\) i.e. such that \(P_i\cap P_j=\emptyset\), \(\forall i,j\) and \(X=\bigcup_1^m\bar{P}_i\). Let \(f:\bigcup_1^m\bar{P}_i\mapsto X\) be such that for every \(i\), \(f_i:=f\mid_{P_i}\) is continuous; since in general \(f\) does not admit a continuous extension to \(X\), let \(D=\bigcup_1^m\partial\bar{P}_i\) be the discontinuity set. Two situations are discussed: when \(f_i\) are homeomorphisms and \(f\) is thus a \textit{piecewise homeomorphism} and when \(f_i\) are isometries and \(f\) is thus a \textit{piecewise isometry}. The paper is concerned with the discontinuity set and its preimage which separate points in \(X\) according to their dynamical behavior. Those whose orbits are arbitrarily close to the discontinuity show such an evolution that might be characterized by a coding map. These maps are studied for piecewise homeomorphisms and piecewise isometries separately.
reviewed by: Vladimir Răsvan
Mathematics Subject Classification ID: 37C05
Mathematics Subject Classification ID: 22C05
zbMATH DE Number: 5704408
zbMATH Keywords: coding map
zbMATH Keywords: piecewise homeomorphisms
zbMATH Keywords: piecewise isometries
author: Miguel Ângelo de Sousa Mendes
MaRDI profile type: Publication
full work available at URL: dcds.2010.27.369
OpenAlex ID: W2041735568
DOI: 10.3934/DCDS.2010.27.369
description: scientific article; zbMATH DE number 5704408