Continuity of the Hausdorff dimension for piecewise monotonic maps
type: Item
instance of: scholarly article
title: Continuity of the Hausdorff dimension for piecewise monotonic maps
zbMATH Open document ID: 0768.28010
DOI: 10.1007/BF02808156
author: Peter Raith
published in: Israel Journal of Mathematics
publication date: 1993-06-29T00:00:00Z
review text: If \(X\) is a finite union of closed intervals, \(T: X\to \mathbb{R}\) a piecewise monotonic map with respect to some partition \(\mathcal Z\) then \((R(T),T)\) with \(R(T)=\bigcap^ \infty_{n=0}\overline{T^{-n}(X)}\) is a dynamical system. The author considers the influence of small perturbations of \(T\) on the set \(R(T)\). He introduces some notions of convergence on the families of all \((X,T,{\mathcal Z})\) and all \((X,T,f,{\mathcal Z})\), where \(f\) is a further piecewise continuous function on \(X\) with respect to \(\mathcal Z\). The principal results are that the pressure function \((X,T,f,{\mathcal Z})\mapsto p(R(t),T,f)\) as well as the Hausdorff dimension function \((X,T,{\mathcal Z})\mapsto \dim(R(t))\) are lower semi-continuous and that jumps of these functions will be bounded in a certain sense. The proofs depend on methods of graph theory.
Mathematics Subject Classification ID: 28D20
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 28D10
Mathematics Subject Classification ID: 37J40
zbMATH DE Number: 219408
zbMATH Keywords: topological pressure
zbMATH Keywords: piecewise monotonic map
zbMATH Keywords: perturbations
zbMATH Keywords: Hausdorff dimension function
reviewed by: Hermann Haase
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 219408