Fractal structure of hyperbolic Lipschitzian dynamical systems
type: Item
instance of: scholarly article
title: Fractal structure of hyperbolic Lipschitzian dynamical systems
zbMATH Open document ID: 1263.37042
DOI: 10.1134/S1061920812010050
published in: Russian Journal of Mathematical Physics
publication date: 2012-10-22T00:00:00Z
review text: In this article, dynamical systems with a Lipschitzian structure are investigated. In the first part of the paper, some properties of hyperbolic Lipschitzian mappings are proved in the terms of the Lipschitzian constants. The main result is the theorem on the semiconjugation of an invertible hyperbolic Lipschitzian dynamical system admiting a pre-Markov partition and the corresponding Markov shift. This result is established in the case there exists some positive constants satisfying the nonlinear system of inequalities. In the second part of the paper, the author suggests a new approach to estimating the Kolmogorov and the Hausdorff dimension of the set of nonwandering points. The proof of these estimations relies on the generalization of the classical theorem on the dimension of the attractor of the matrix iteration system. The generalization is that the contracting condition is replaced by a weaker assumption of contraction along the simple circle.
Mathematics Subject Classification ID: 37C45
Mathematics Subject Classification ID: 37B10
Mathematics Subject Classification ID: 37D99
Mathematics Subject Classification ID: 26B35
zbMATH DE Number: 6097099
zbMATH Keywords: Lipschitzian mapping
zbMATH Keywords: nonwandering points
zbMATH Keywords: Markov partition
zbMATH Keywords: topological Markov chain
zbMATH Keywords: attractor
zbMATH Keywords: iteration system
zbMATH Keywords: Kolmogorov dimension
zbMATH Keywords: Hausdorff dimension
author: L. V. Lokutsievskiy
reviewed by: Ivan Podvigin
MaRDI profile type: Publication
cites work: Q4840149
cites work: Q4285325
cites work: Q4376268
cites work: Q4733149
description: scientific article; zbMATH DE number 6097099