Measures and their dimension spectrums for cookie-cutter sets in \(\mathbb{R}^d\)
type: Item
instance of: scholarly article
title: Measures and their dimension spectrums for cookie-cutter sets in \(\mathbb{R}^d\)
zbMATH Open document ID: 0970.28004
DOI: 10.1007/BF02670959
published in: Acta Mathematicae Applicatae Sinica. English Series
publication date: 2001-02-04T00:00:00Z
review text: Cookie-cutter sets have been intensively studied either as fractal repellers of smooth expansive dynamical systems (cookie-cutters) or as invariant sets of related nonlinear contracting iterated function systems. In this paper, fractal geometric theory of cookie-cutter sets in \(n\)-dimensional Euclidean space is derived assuming certain conditions. These conditions basically ensure that the so-called bounded distortion property holds so that the methods developed in [\textit{T. Bedford}, in: Fractal geometry and analysis, NATO ASI Ser., Ser. C 346, 1-44 (1991; Zbl 0741.58011)] for the one-dimensional case apply in higher dimensions. Specifically, the paper extends to its higher dimensional setting: (i) Bowen's pressure formula for the Hausdorff dimension of the invariant set, (ii) existence and natural properties, including multifractal decomposition, of self-similar, conformal and Gibbs measures, and (iii) existence of constant order-two density. The situation considered in the paper includes self-similar sets and self-conformal sets as particular cases, for which most of the results in the paper are well-known [\textit{K. J. Falconer}, ``Techniques in fractal geometry'' (1998; Zbl 0869.28003)]. In [\textit{L. M. Barreira}, Ergodic Theory Dyn. Syst. 16, No. 5, 871-927 (1996; Zbl 0862.58042)], dimension theory for hyperbolic sets is developed in a broad setting that includes more general classes of non-conformal repellers.
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 37C45
zbMATH DE Number: 1462489
zbMATH Keywords: cookie-cutter sets
zbMATH Keywords: fractals
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: multifractals
zbMATH Keywords: Gibbs measures
zbMATH Keywords: order-two densities
zbMATH Keywords: self-similar sets
zbMATH Keywords: self-conformal sets
author: Jin-Rong Liang
author: Zu-Guo Yu
author: Fuyao Ren
MaRDI profile type: Publication
cites work: Q3731958
cites work: Multifractal decompositions of Moran fractals
cites work: Multifractal Decompositions of Digraph Recursive Fractals
cites work: A multifractal formalism
cites work: The Hausdorff dimension and measure of the generalized Moran fractals and Fourier series
cites work: Multifractal formalism for infinite multinomial measures
cites work: Self-similarity in harmonic analysis
cites work: Q4040874
cites work: Ergodic theory. Introductory lectures
cites work: Equilibrium states and the ergodic theory of Anosov diffeomorphisms
cites work: A Variational Principle for the Pressure of Continuous Transformations
cites work: On the Existence of Conformal Measures
cites work: Fourier asymptotics of fractal measures
cites work: Q4865022
cites work: Q3714302
cites work: Wavelet transforms and order-two densities of fractals
full work available at URL: https://doi.org/10.1007/bf02670959
OpenAlex ID: W108790492
description: scientific article; zbMATH DE number 1462489