Dynamical systems, measures, and fractals via domain theory
type: Item
instance of: scholarly article
title: Dynamical systems, measures, and fractals via domain theory
zbMATH Open document ID: 0834.58029
author: Abbas Edalat
published in: Information and Computation
publication date: 1996-04-01T00:00:00Z
review text: We introduce domain theory in dynamical systems, iterated function systems (fractals), and measure theory. For a discrete dynamical system given by the action of a continuous map \(f: X\to X\) on a metric space \(X\), we study the extended dynamical systems \((VX, Vf)\), \((UX, Uf)\), and \((LX, Lf)\), where \(V\), \(U\), and \(L\) are respectively the Vietoris hyperspace, the upper hyperspace, and the lower hyperspace functors. We show that if \((X, f)\) is chaotic, then so is \((UX, Uf)\). When \(X\) is locally compact, \(UX\) is a continuous bounded complete dcpo. If \(X\) is second countable as well, then \(UX\) will be \(\omega\)-continuous and can be given an effective structure. We show how strange attractors, attractors of iterated function systems (fractals) and Julia sets are obtained effectively as fixed points of deterministic functions on \(UX\) or fixed points of non-deterministic functions on \(CUX\) where \(C\) is the convex (Plotkin) power domain. We also show that the set, \(M(X)\), of finite Borel measures on \(X\) can be embedded in \(PUX\), where \(P\) is the probabilistic power domain. This provides an effective framework for measure theory. We then prove that the invariant measure of an hyperbolic iterated function system with probabilities can be obtained as the unique fixed point of an associated continuous function on \(PUX\).
Mathematics Subject Classification ID: 37D45
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 68Q55
Mathematics Subject Classification ID: 37A99
zbMATH DE Number: 797316
zbMATH Keywords: fractals
zbMATH Keywords: domain theory
zbMATH Keywords: dynamical systems
zbMATH Keywords: iterated function systems
zbMATH Keywords: measure theory
zbMATH Keywords: Julia sets
MaRDI profile type: Publication
full work available at URL: inco.1995.1096
OpenAlex ID: W3096424428
DOI: 10.1006/INCO.1995.1096
description: scientific article; zbMATH DE number 797316