Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages.
type: Item
instance of: scholarly article
title: Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages.
zbMATH Open document ID: 1035.37025
author: L. Olsen
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2004-01-26T00:00:00Z
review text: The aim of this paper is to develop a unifying multifractal framework, based on the so-called deformations of empirical measures, which allows significant extensions of already known results in multifractal analysis of both dynamical systems and fractal measures. One considers a finite directed graph \((V,E)\). To each vertex \(i\in V\) one associates the set \(\Sigma_i\) of all infinite paths in \((V,E)\) starting at \(i\), and the set \(\Sigma\) of all infinite paths in \(V\). \(S:\Sigma\to\Sigma\) is the shift map, and \({\mathcal P}(\Sigma)\) the family of probability measures on \(\Sigma\). For \(n\in \mathbb N\), \(L_{n}:\Sigma\to{\mathcal P}(\Sigma)\) is the \(n\)th empirical measure, defined by: \[ L_{n}\omega= {{1}\over{n}} \sum_{k=0}^{n-1} \delta_{S^{k}\omega}, \] where \(\delta_{x}\) is the Dirac measure at \(x\). A continuous deformation of \(L_{n}\) is a pair \((X,\Xi)\), where \(X\) is a metric space, and \[ \Xi:{\mathcal P}(\Sigma)\to X \] is a continuous map with respect to the weak topology. The multifractal spectrum of this deformation is defined by: \[ f_{i}(t)=\text{dim} \left(\pi_{i}\{\omega\in \Sigma_{i}\mid \lim_{n\to\infty}\Xi L_{n}\omega=t\}\right), \] where \(\pi_{i}:\Sigma_{i}\to \mathbb R^{d}\) is the natural projection. For different choices of the deformation one obtains different multifractal spectra. As examples of deformations of empirical measures are given: the spectrum of ergodic averages, the multifractal spectrum of the self-conformal measures \(\mu_i\), associated with a directed graph iterated self-conformal function system with probabilities, the spectrum of local entropies of \(\mu_i\), the spectrum of local Lyapunov exponents. The main result (Theorem 5) gives lower and upper bounds for the multifractal spectra for arbitrary nonlinear continuous deformations. There are also given necessary conditions ensuring the equality in Theorem 5. These results lead to an abstract notion of relative or conditional multifractal spectra.…
reviewed by: Emilia Petrisor
Mathematics Subject Classification ID: 37D35
Mathematics Subject Classification ID: 37C45
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 2032770
zbMATH Keywords: deformed measures
zbMATH Keywords: multifractal
zbMATH Keywords: divergence point
zbMATH Keywords: local Lyapunov exponents
zbMATH Keywords: local entropies
MaRDI profile type: Publication
full work available at URL: j.matpur.2003.09.007
OpenAlex ID: W1989468930
cites work: Random Self-Similar Multifractals
cites work: Variational principles and mixed multifractal spectra
cites work: Sets of ``non-typical'' points have full topological entropy and full Hausdorff dimension
cites work: Q3974721
cites work: Generalized fractal dimensions: equivalences and basic properties.
cites work: Equilibrium states and the ergodic theory of Anosov diffeomorphisms
cites work: Billingsley dimension in probability spaces
cites work: Multifractal decompositions of Moran fractals
cites work: Relative multifractal analysis
cites work: Q4853135
cites work: The pointwise dimension of self-similar measures
cites work: Local properties of self-similar measures
cites work: Ergodic theory on compact spaces
cites work: Q3134548
cites work: Q3681702
cites work: Large deviations for \(\mathbb{Z}^ d\)-actions
cites work: Multifractal Decompositions of Digraph Recursive Fractals
cites work: Q3714302
cites work: Q3995301
cites work: Techniques in fractal geometry
cites work: RECURRENCE, DIMENSION AND ENTROPY
cites work: Fractal measures and their singularities: The characterization of strange sets
cites work: Thermodynamic formalism and multifractal analysis of conformal infinite iterated function systems
cites work: The infinite number of generalized dimensions of fractals and strange attractors
cites work: Q3731958
cites work: Multifractal Formalism for Functions Part I: Results Valid For All Functions
cites work: Multifractal Formalism for Functions Part II: Self-Similar Functions
cites work: Singularity spectrum for recurrent IFS attractors
cites work: Large Deviations in Dynamical Systems and Stochastic Processes
cites work: Q4861724
cites work: Multifractal analysis of Choquet capacities
cites work: Q4945265
cites work: Analyse multifractale de fonctions continues
cites work: Structure multifractale d'une dynamique non expansive définie sur un ensemble de Cantor
cites work: Q4301606
cites work: Self-affine multifractal Sierpiński sponges in \(\mathbb{R}^d\)
cites work: Q4945263
cites work: DISTRIBUTION OF DIGITS IN INTEGERS: BESICOVITCH–EGGLESTON SUBSETS OF ${\mathbb N}$
cites work: Small sets of divergence points are dimensionless
cites work: Applications of multifractal divergence points to sets of numbers defined by their $N$ -adic expansion
cites work: Applications of multifractal divergence points to some sets of \(d\)-tuples of numbers defined by their \(N\)-adic expansion
cites work: Mixed divergence points of self-similar measures
cites work: NORMAL AND NON-NORMAL POINTS OF SELF-SIMILAR SETS AND DIVERGENCE POINTS OF SELF-SIMILAR MEASURES
cites work: Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. II: non-linearity, divergence points and Banach space valued spectra
cites work: Self-conformal multifractal measures
cites work: Multifractal analysis of Lyapunov exponent for continued fraction and Manneville-Pomeau transformations and applications to diophantine approximation
cites work: The singularity spectrumf(α) for cookie-cutters
cites work: Separation Properties for Self-Similar Sets
cites work: Non-negative matrices and Markov chains. 2nd ed
cites work: Multifractal analysis of local entropies for expansive homeomorphisms with specification
cites work: Asymptotic probabilities and differential equations
cites work: Q3931654
cites work: Q4384907
DOI: 10.1016/J.MATPUR.2003.09.007
description: scientific article; zbMATH DE number 2032770