The box dimension of completely invariant subsets for expanding piecewise monotonic transformations
type: Item
instance of: scholarly article
title: The box dimension of completely invariant subsets for expanding piecewise monotonic transformations
zbMATH Open document ID: 0845.28004
DOI: 10.1007/BF01298950
author: Franz Hofbauer
published in: Monatshefte für Mathematik
publication date: 1996-09-11T00:00:00Z
full work available at URL: https://eudml.org/doc/178721
review text: Let \(T\) be an expanding piecewise monotonic transformation of the unit interval, having a regular derivative \(T'\), and let \(A\) be a closed completely invariant subset of \(([0, 1], T)\) which is topologically transitive with positive entropy. The author shows that the unique zero \(z_A\) of the pressure function \(t\mapsto p(T|A,- t\log |T'|)\) is equal to the box, the Hausdorff as well as the packing dimension of \(A\). The assumption that \(T\) is expanding cannot be weakened. There are some analogues to Julia sets.
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 26A18
Mathematics Subject Classification ID: 28D20
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 874080
zbMATH Keywords: fractal dimension
zbMATH Keywords: box dimension
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: expanding piecewise monotonic transformation
zbMATH Keywords: entropy
zbMATH Keywords: packing dimension
reviewed by: Hermann Haase
MaRDI profile type: Publication
cites work: Q3995301
cites work: Hausdorff and conformal measures for expanding piecewise monotonic maps of the interval
cites work: Local dimension for piecewise monotonic maps on the interval
cites work: Fractal Properties of Invariant Subsets for Piecewise Monotonic Maps on the Interval
cites work: Hausdorff dimension for piecewise monotonic maps
cites work: Q4233869
cites work: Q3931654
description: scientific article; zbMATH DE number 874080