Q2054458
type: Item
instance of: scholarly article
title: Non-invertible dynamical systems. Volume 2. Finer thermodynamic formalism -- distance expanding maps and countable state subshifts of finite type, conformal GDMSs, Lasota-Yorke maps and fractal geometry
zbMATH Open document ID: 1505.37003
DOI: 10.1515/9783110702699
author: Mariusz Urbański
author: Mario Roy
author: Sara Munday
published in: De Gruyter Expositions in Mathematics
publication date: 2021-12-02T00:00:00Z
review text: This is the second volume in a series of three on topological dynamical systems (for a review of the first volume, see [\textit{M. Urbański} et al., Non-Invertible dynamical systems. Volume 1. Ergodic theory -- finite and infinite, thermodynamic formalism, symbolic dynamics and distance expanding maps. Berlin: De Gruyter (2022; Zbl 1493.37003)]). The first volume developed the general thermodynamic formalism for such systems, with special emphasis on those that involve continuous self-maps of a compact metrizable space with continuous potentials. This second volume is more focused on topical and more advanced aspects. It continues with the thermodynamic formalism and its geometric and dynamical applications, but it narrows its focus to specific systems and classes of systems with specific potentials. This volume begins with an introduction to the concept of Gibbs state as a core of the thermodynamic formalism and a basic tool for studying the fractal geometry of invariant sets that are defined by dynamics. The Perron-Frobenius operator, also commonly called the transfer operator, is also introduced as a primary tool of the thermodynamic formalism, and designed to construct Gibbs states. These concepts are discussed in the context of open, distance-expanding maps under continuous potentials. Next the authors take up Lasota-Yorke maps, generally piecewise differentiable expanding maps of a compact interval with finitely many pieces. Their transfer operator is used to establish existence of an invariant measure absolutely continuous with respect to Lebesgue measure. With some additional assumptions, the authors establish exponential decay of correlations with respect to that invariant measure. The third section departs from considerations of dynamics to describe various ways of evaluating the size of fractal sets, particularly fractal measures and fractal dimensions. The authors focus on geometric measures like packing and Hausdorff and the dimensions that they induce.…
Mathematics Subject Classification ID: 37-02
Mathematics Subject Classification ID: 37B05
Mathematics Subject Classification ID: 37D20
Mathematics Subject Classification ID: 37D25
Mathematics Subject Classification ID: 37D30
Mathematics Subject Classification ID: 37D35
Mathematics Subject Classification ID: 37E05
Mathematics Subject Classification ID: 37E15
Mathematics Subject Classification ID: 37E25
Mathematics Subject Classification ID: 37C45
Mathematics Subject Classification ID: 37C30
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 7438186
zbMATH Keywords: dynamical systems
zbMATH Keywords: non-invertible dynamics
zbMATH Keywords: Lasota-Yorke-maps
zbMATH Keywords: conformal graph directed Markov systems
reviewed by: William J. Satzer Jr.
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1515/9783110702699
OpenAlex ID: W4281481184
label: Non-invertible dynamical systems. Volume 2. Finer thermodynamic formalism -- distance expanding maps and countable state subshifts of finite type, conformal GDMSs, Lasota-Yorke maps and fractal geometry
description: scientific article; zbMATH DE number 7438186