Multifractal analysis of infinite products
type: Item
instance of: scholarly article
title: Multifractal analysis of infinite products
zbMATH Open document ID: 0928.60086
DOI: 10.1007/BF02183625
published in: Journal of Statistical Physics
publication date: 2000-01-02T00:00:00Z
review text: A multifractal formalism is formulated for a family of measures called infinite products, which generalize the Gibbs measures in the one-dimensional lattice gas model. If \(\{\lambda_n\}\) is a lacunary sequence of positive integers and \(\{g_n\}\) a sequence of positive periodic, continuous functions on the circle, then the infinite products are the weak limit points of the probability measures \(Z_n^{-1} P_n(t) dt\), where \(P_n(t)=\prod_{k=0}^{n-1} g_k(\lambda_k t)\) and \(Z_n\) is the normalizing factor. Under given conditions the infinite product is well defined and there is a family of Gibbs measures associated to it. The singularity spectrum formula describes the Hausdorff or packing dimension of the set of points of local dimension \(\alpha\) as the Legendre transform at \(\alpha\) of the so-called \(\tau\)-function. The main result of the paper shows that for infinite products under some conditions a weak form of the singularity spectrum formula holds. One of these conditions, the existence of the \(\varphi\)-function, is verified in important special cases, the others are explicit in \(\{\lambda_n\}\) and \(\{g_n\}\). It is also shown, that if the \(\tau\)-function is differentiable, the full singularity spectrum formula holds. These results imply new results on the multifractal analysis of Riesz products and the methods of the paper also extend to \(G\)-measures.
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B31
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 1279779
zbMATH Keywords: Gibbs measure
zbMATH Keywords: Riesz product
zbMATH Keywords: infinite product
zbMATH Keywords: pressure function
zbMATH Keywords: multifractal analysis
zbMATH Keywords: multifractal formalism
author: Ai-Hua Fan
MaRDI profile type: Publication
cites work: The entropy function for characteristic exponents
cites work: Equilibrium states and the ergodic theory of Anosov diffeomorphisms
cites work: Odometer actions on G-measures
cites work: On the multifractal analysis of measures
cites work: Multifractal decompositions of Moran fractals
cites work: The dimension spectrum of some dynamical systems
cites work: Large deviations for a general class of random vectors
cites work: Sur les dimensions de mesures
cites work: On uniqueness of G-measures and g-measures
cites work: ERGODICITY, UNIDIMENSIONALITY AND MULTIFRACTALITY OF SELF-SIMILAR MEASURES
cites work: Q4865381
cites work: Fractal measures and their singularities: The characterization of strange sets
cites work: Strongly mixing g-measures
cites work: Fractal measures and mean \(p\)-variations
cites work: Multifractal measures and a weak separation condition
cites work: Mean quadratic variations and Fourier asymptotics of self-similar measures
cites work: Q4328334
cites work: A multifractal formalism
cites work: Étude de quelques propriétés des produits de Riesz
cites work: The singularity spectrumf(α) for cookie-cutters
cites work: Convex Analysis
cites work: Q4185090
cites work: Q4287683
cites work: DIMENSIONS IN A SEPARABLE METRIC SPACE
cites work: Two definitions of fractional dimension
cites work: On multipliers preserving convergence of trigonometric series almost everywhere
description: scientific article; zbMATH DE number 1279779