Ergodic theory of expanding Thurston maps
type: Item
instance of: scholarly article
title: Ergodic theory of expanding Thurston maps
zbMATH Open document ID: 1380.37003
DOI: 10.2991/978-94-6239-174-1
published in: Atlantis Studies in Dynamical Systems
publication date: 2016-03-23T00:00:00Z
full work available at URL: 33g1m263
Mathematics Subject Classification ID: 37-02
Mathematics Subject Classification ID: 37D35
Mathematics Subject Classification ID: 37Axx
zbMATH DE Number: 6560003
zbMATH Keywords: Thurston map
zbMATH Keywords: measure of maximal entropy
zbMATH Keywords: thermodynamical formalism
zbMATH Keywords: equilibrium state
zbMATH Keywords: large deviation principle
author: Zhiqiang Li
reviewed by: Ivan Podvigin
MaRDI profile type: Publication
OpenAlex ID: W1638782246
review text: Let \(S^2\) be 2-sphere and \({f:S^2\to S^2}\) is a branched covering map on \(S^2,\) i.e., it is a continuous map which for all \({x\in S^2}\) locally satisfies the equation \({(\eta\circ f\circ\varphi^{-1})(z)=z^d}\) for some \(d\in\mathbb{N}\) and orientation-preserving homeomorphisms \({\eta,\varphi:S^2\to\widehat{\mathbb{C}}}.\) The integer \(d\) is a local degree of \(f\) at \(x\), denoted by \({\operatorname{deg}_f(x)}.\) If \({\operatorname{deg}_f(x)\geq2}\) then \(x\) is called critical and the set of such points is denoted by \({\operatorname{crit}f}.\)NEWLINENEWLINEThe monograph is devoted to the study of ergodic properties of some branched covering maps called expanding Thurston maps. A Thurston map is a branched covering map for which NEWLINE\[NEWLINE \operatorname{card}\bigcup_{n=1}^\infty f^n(\operatorname{crit}f)<\infty NEWLINE\]NEWLINE and topological degree of \(f\) NEWLINE\[NEWLINE \operatorname{deg}f:=\sum_{x\in f^{-1}(y)}\operatorname{deg}_f(x)\geq2. NEWLINE\]NEWLINE A Thurston map induces the so-called cell decompositions \(\mathbf{D}^n\) on \(S^2\) for all \({n\in\mathbb{N}}.\) If the diameter (with respect to some metric) of the cells of \(\mathbf{D}^n\) tends to zero as \({n\to\infty}\) then the Thurston map is called expanding. The author uses the so-called visual metric, which gives a good control over size of the cells.NEWLINENEWLINEThe monograph focuses on the properties of some invariant measures of expanding Thurston maps. The author considers the measure of maximal entropy and investigates its various properties, in particular the equidistribution of preimage points, periodic and preperiodic points. Then the author applies the thermodynamical formalism to study equilibrium states. Some equidistribution results are also obtained.NEWLINENEWLINEThe book is interesting, contains meaningful results on ergodic theory of expanding Thurston maps and is an excellent supplement to the book [\textit{M. Bonk} and \textit{D. Meyer}, Expanding Thur…
description: scientific article; zbMATH DE number 6560003