On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann-Ruelle conjecture
type: Item
instance of: scholarly article
title: On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann-Ruelle conjecture
zbMATH Open document ID: 0878.28006
DOI: 10.1007/BF02506387
author: Howard Weiss
author: Yakov Pesin
published in: Communications in Mathematical Physics
publication date: 1998-01-08T00:00:00Z
review text: This paper unifies and extends many of known results on the dimension of deterministic and random Cantor-like sets in \(\mathbb{R}^n\). The authors verify the Eckmann-Ruelle conjecture for equilibrium measures for Hölder continuous conformal expanding maps and conformal Axiom \(A^\#\) homeomorphism. They also show that the non-conformal Hölder continuous version of the conjecture is false. The Cantor-like sets will be constructed by arbitrary symbolic dynamical systems. The original idea of Moran will be extended to much broader classes of geometric constructions. The basic Moran-like construction starts with a compact shift invariant subset \(Q\) of an infinite product space of \(p\) symbols. The basic geometric sets \(\Delta_{i_1,\dots,i_n}\) in \(\mathbb{R}^n\), where \((i_1,\dots,i_n,i_{n+1}',\dots)\in Q\), have an inner and an outer closed ball, such that the inner balls for fixed \(n\) are disjoint, and the radii of both types equal to the product of the ratio coefficients associated to the symbols \(i_1,\dots,i_n\) (up to two positive constants).
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 37E99
zbMATH DE Number: 1002420
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: box dimension
zbMATH Keywords: Cantor-like sets
zbMATH Keywords: Eckmann-Ruelle conjecture
zbMATH Keywords: equilibrium measures
zbMATH Keywords: Moran-like construction
Wikidata QID: Q122998900
reviewed by: Hermann Haase
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02506387
OpenAlex ID: W2000309080
description: scientific article; zbMATH DE number 1002420