Ergodic theory of expanding Thurston maps
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 Thurston maps. Providence, RI: American Mathematical Society (AMS) (2017; Zbl 1430.37001)]. The book will be useful to specialists in ergodic theory as well as to beginning researchers.
scientific article; zbMATH DE number 6560003