Ergodic limits on the conformal repellers
type: Item
instance of: scholarly article
title: Ergodic limits on the conformal repellers
zbMATH Open document ID: 1033.37017
author: De-Jun Feng
author: Ka-Sing Lau
author: Jun Wu
published in: Advances in Mathematics
publication date: 2002-09-12T00:00:00Z
review text: Let \(M\) be a smooth Riemannian manifold and \(g:M\rightarrow M\) be a \(C^{1+\delta}\)-conformal map. Let \(\Phi\) be a continuous function defined on a \(g\)-invariant compact subset \(J\) of \(M\) with values in \(\mathbb{R}^{d}.\) For any \(x\in J,\) the ergodic limit, if exists, is defined by \[ \alpha(x)=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{j=0}^{n-1}\Phi(g^{j}x). \] For \(\alpha\in L_{\Phi},\) let \[ E(\alpha)=\left\{ x\in J:\alpha(x)=\alpha\right\} , \] where \[ L_{\Phi}=\left\{ \alpha : \alpha=\alpha(x)\text{ for some }x\in J\right\} . \] The main purpose of the paper is to examine the size of the set \(E(\alpha)\) as well as the size of the set of points such that the limits defining \(\alpha(x)\) do not exist. The sizes of these sets are described in terms of the Hausdorff dimension \(\dim_{H}\) and the packing dimension \(\dim_{P}.\)
Mathematics Subject Classification ID: 37C70
Mathematics Subject Classification ID: 37A25
Mathematics Subject Classification ID: 37F15
Mathematics Subject Classification ID: 37C45
zbMATH DE Number: 1800237
zbMATH Keywords: repeller
zbMATH Keywords: expanding map
zbMATH Keywords: conformal topologically mixing map
zbMATH Keywords: ergodic limit
zbMATH Keywords: entropy
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: Markov measure
reviewed by: Yuriy V. Rogovchenko
MaRDI profile type: Publication
full work available at URL: aima.2001.2054
OpenAlex ID: W2169469642
cites work: On the sum of digits of real numbers represented in the dyadic system. (On sets of fractional dimensions II.)
cites work: Q5515914
cites work: Q4026037
cites work: Invariant sets with zero measure and full Hausdorff dimension
cites work: Sets of ``non-typical'' points have full topological entropy and full Hausdorff dimension
cites work: Equilibrium states and the ergodic theory of Anosov diffeomorphisms
cites work: Q3316247
cites work: Multifractal decompositions of Moran fractals
cites work: Multifractal Decompositions of Digraph Recursive Fractals
cites work: THE FRACTIONAL DIMENSION OF A SET DEFINED BY DECIMAL PROPERTIES
cites work: Q3995301
cites work: Sur les dimensions de mesures
cites work: On the distribution of long-term time averages on symbolic space
cites work: RECURRENCE, DIMENSION AND ENTROPY
cites work: Asymptotic behavior of multiperiodic functions \(G(x)=\prod_{n=1}^\infty g(x/2^n)\)
cites work: Iterated function system and Ruelle operator
cites work: On the structures and dimensions of Moran sets
cites work: Q4724859
cites work: Q4328334
cites work: Multifractal analysis in symbolic dynamics and distribution of pointwise dimension forg-measures
cites work: Topological pressure and the variational principle for noncompact sets
cites work: A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions
cites work: Net measure properties of Moran sets and applications
cites work: Q4185090
cites work: Q3931654
cites work: The Lyapunov spectrum for conformal expanding maps and Axiom-A surface diffeomorphisms
DOI: 10.1006/AIMA.2001.2054
description: scientific article; zbMATH DE number 1800237