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
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. In their first volume the authors had explored the class or expending repellers, and here, in the latter part of the book, they restrict their attention to conformal expanding repellers. One aspect of their work is to carry out a multifractal analysis of the Hausdorff dimension of the level sets of Gibbs states for any Hölder continuous potential. The final five chapters consider countable state thermodynamic formalism and the theory of conformal graph directed Markov systems. The techniques described here provide powerful methods for constructing and examining the dynamic and geometric properties of fractal sets. The book could be used as a text for a graduate course or seminar in advanced dynamical systems theory. Each chapter in the book has a modest set of exercises.
scientific article; zbMATH DE number 7438186