This book offers a clear approach to the dynamics of Thurston maps under iteration. The topics in the book bring the subject closer to the reader and provide a very useful guide to those who want to do research in the area. For the readers the prerequisites include complex analysis, plane topology and topology of surfaces.
The main focus of the book is given to Thurston maps, namely (non-homeomorphic) branched covering maps \(f: \mathbb{S}^2 \to \mathbb{S}^2\) that are postcritically finite. The authors assume that the maps \(f\) are expanding in a suitable sense. Such maps give rise to a fractal geometry which is represented by a class of visual metrics that are associated with the maps.
In the introductory chapter the proposal of the authors is to give some guidance for the intuition of the reader. The authors present a specific Lattés map which motivates several concepts important for a general Thurston map. The last section contains all the examples of Thurston maps that are considered in the book.
Chapter 2 is related to main object of study in the book with the following topics: branched covering maps, definition of Thurston maps, definition of expansions, Thurston equivalence, orbifolds associated with a Thurston map and Thurston characterization of rational maps.
Chapter 3 contains the definition and the characterization of Lattés maps. Here there is a discussion of a large class of Thurston maps, namely Lattés maps, and a related class called Lattés-type maps which are quotients of torus endomorphisms.
Chapter 4 is related to quasiconformal geometry, Gromov hyperbolicity, Gromov hyperbolic groups and Cannon’s conjecture and quasispheres.
Chapter 5 contains some technical aspects such as cell decompositions and their relation with Thurston maps.
In Chapter 6 the notion of expansion of Thurston maps is studied in depth.
In Chapter 7 Thurston maps with two or three postcritical points are considered. Every such map is equivalent to a rational function. It is also considered a Thurston map with an associated parabolic orbifold of signature \(( \infty, \infty)\) or \((2, 2, \infty)\).
Chapter 8 introduces a visual metrics for expanding Thurston maps with its properties and characterization.
Chapter 9 deals with the study of symbolic dynamics for expanding Thurston maps and Chapter 10 deals with the study and properties of tile graphs.
Chapter 11 is devoted to the Thurston equivalence defined in terms of certain isotopies. The chapter contains the following topics: conjugation of equivalent expanding Thurston maps, isotopies of Jordan curves and isotopies of cell decomposition.
Chapter 12 contains a summary of facts related to Thurston maps with invariant curves and the associated cell decompositions and two-tile subdivision rules.
Chapter 13 contains the study of the general problem occurring when a Thurston map passes to another Thurston map on a quotient of \(\mathbb{S}^2\) induced by an equivalence relation.
In Chapter 14 the key concept of combinatorial expansion is used to understand when the Thurston map realizing a subdivision rule can be expanding.
In Chapter 15 some existence and uniqueness results for invariant curves of an expanding Thurston map are proved. The chapter contains an iterative procedure to obtain invariant curves. It is proved that a Jordan curve is a quasicircle if it is invariant for an expanding Thurston map.
Chapter 16 introduces a combinatorial expansion factor to prove the main result in the chapter (Theorem 16.3) which relates the combinatorial expansion factor to expansion factors of visual metrics.
In Chapter 17 the measure of maximal entropy of an expanding Thurston map is investigated. This chapter contains a review on measure-theoretic dynamics and the construction of the measure of maximal entropy (given a fix expanding Thurston map).
Chapter 18 contains the geometry of the visual sphere of an expanding Thurston map. The topics in the chapter are linear local connectedness, doubling and Ahlfors regularity and quasisymmetric and rational Thurston maps.
In Chapter 19 the measure-theoretic properties of rational expanding Thurston maps are studied. Also, some results crucial for the characterization of Lattés maps are discussed. The topics in Chapter 19 are: Jacobian of measurable maps, ergodicity of the Lebesgue measure and absolute continuous invariant measures.
In Chapter 20 the authors formulate criteria for an expanding Thurston map to be topologically conjugate to a Lattés map in terms of the existence of visual metrics with special properties. It is shown that Lattés maps can be characterized in terms of their combinatorial expansion behavior.
Chapter 21 contains an outlook on further results an open problems related to the subject of the book.