Iterated function systems and the code space
type: Item
instance of: scholarly article
title: Iterated function systems and the code space
zbMATH Open document ID: 1021.37011
DOI: 10.1016/S0166-8641(01)00134-1
published in: Topology and its Applications
publication date: 2002-08-22T00:00:00Z
review text: Let \(\{1,2,\ldots, N\}\) be a finite alphabet and \(\Sigma=\prod_{i=1}^\infty\{1,2,\ldots, N\}\) the associated code space. This space is the domain of the address map \(g\) associated to an iterated function system (IFS) of similitudes \(\{f_{1}, f_{2},\ldots,f_{N}\}\), with \(f_{i}:\mathbb{R}^{n}\to\mathbb{R}^{n}\) of similarity factor \(0<c_{i}<1\), \(i=1,\dots,N\). Namely, if \(K\) is the selfsimilar set associated to the IFS, then \(g:\Sigma\to K\) is defined by: \[ g(i_{1}, i_{2},\ldots, i_{n},\ldots)=\lim_{n\to\infty}f_{i_{1}}\circ\cdots\circ f_{i_{n}}. \] In this paper the authors define an ultrametric on the code space and establish a natural connection between the corresponding metric space and the invariant set \(K\) of the IFS having as similarity factors the constants used in the definition of the ultrametric. Independently on an IFS one considers a set of numbers \(0<c_{i}<1\), \(i=1,\dots,N\). Then the ultrametric \(\rho\) defined on the code space \(\Sigma\) is given by \(\rho(i_{j},k_{j})=c_{i_{1}}\cdots c_{i_{n}}\), if the two sequences agree up to index \(n\) and disagree at index \(n+1\); \(\rho(i_{j},k_{j})=0\) if the two sequences are identical, and \(\rho(i_{j},k_{j})=1\) if they differ in the first element of the sequence. One of the results of the paper states that if the constants \(c_{i}\) satisfy \(0<c_{i}<{1\over{2}}\), \(i\in\{1,2\ldots,N\}\), then there is a bi-Lipschitz embedding \(g\) of the metric space \((\Sigma,\rho)\) in some \(\mathbb{R}^{n}\), where \(n\) is any integer satisfying \(2^{n}\geq N\). Moreover the embedding \(g\) is such that \(g(\Sigma)\) is a selfsimilar set defined by an IFS of \(N\) similitudes, of similarity factors \(\{c_{1},\ldots, c_{N}\}\), and satisfying the open set condition. A second result consists in a formula for the Hausdorff dimension of a subshift of finite type, \(\Sigma_{B}\), of the metric space \((\Sigma,\rho)\) (\(B=(b_{ij})\) is the transition matrix defining the subshift). Namely, it is s…
reviewed by: Emilia Petrisor
Mathematics Subject Classification ID: 37B40
Mathematics Subject Classification ID: 37B10
Mathematics Subject Classification ID: 37C45
Mathematics Subject Classification ID: 37B25
Mathematics Subject Classification ID: 37D05
Mathematics Subject Classification ID: 37A05
zbMATH DE Number: 1787550
zbMATH Keywords: code space
zbMATH Keywords: subshift of finite type
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: selfsimilar set
zbMATH Keywords: subselfsimilar set
zbMATH Keywords: topological entropy
zbMATH Keywords: Lyapunov exponent
author: L. Block
author: J. Keesling
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1787550