Dynamics of piecewise isometries
type: Item
instance of: scholarly article
title: Dynamics of piecewise isometries
zbMATH Open document ID: 0964.37009
author: Arek Goetz
published in: Illinois Journal of Mathematics
publication date: 2001-07-16T00:00:00Z
review text: The subject of the paper is Euclidean piecewise isometric dynamical systems (p.i.d.s.). A piecewise isometry is a pair \((T,\mathcal P)\), where \(T: X \to X\) is a map such that its restriction to \(P_i\), \(i=0,\dots,r-1\) is a Euclidean isometry. Here \(X\) is a subset of \(\mathbb{R}^N\) and \({\mathcal P} = \{ P_0,\dots, P_{r-1}\}\) \((r>1)\) is a finite partition of \(X\). Such systems are direct generalization of interval exchange transformations to non-invertible, higher-dimensional maps. Considering geometrical properties of p.i.d.s. and symbolic dynamics the author describes the relation between the growth of the associated semigroup of isometries and the growth of symbolic words. The notion of entropy for piecewise isometry is introduced and expressed in terms of the growth of symbolic words. Furthermore, the author studies necessary conditions for a p.i.d.s. to generate all possible finite words. Finally, some remarks on the interplay between symbolic codings, periodic points, and the geometry of cells (i.e. sets following the same coding pattern) are given. A number of results included are generalizations of results for interval exchangers.
reviewed by: S. Yanchuk
Mathematics Subject Classification ID: 37B10
Mathematics Subject Classification ID: 37B40
Mathematics Subject Classification ID: 37F99
Mathematics Subject Classification ID: 52A07
zbMATH DE Number: 1525073
zbMATH Keywords: piecewise isometries
zbMATH Keywords: symbolic growth
zbMATH Keywords: symbolic dynamics
zbMATH Keywords: encoding
zbMATH Keywords: entropy
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1525073