CiNii Books - Expanding Thurston maps

This monograph is devoted to the study of the dynamics of expanding Thurston maps under iteration. A Thurston map is a branched covering map on a two-dimensional topological sphere such that each critical point of the map has a finite orbit under iteration. A Thurston map is called expanding if, roughly speaking, preimages of a fine open cover of the underlying sphere under iterates of the map become finer and finer as the order of the iterate increases. Every expanding Thurston map gives rise to a fractal space, called its visual sphere. Many dynamical properties of the map are encoded in the geometry of this visual sphere. For example, an expanding Thurston map is topologically conjugate to a rational map if and only if its visual sphere is quasisymmetrically equivalent to the Riemann sphere. This relation between dynamics and fractal geometry is the main focus for the investigations in this work.

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Introduction Thurston maps Lattes maps Quasiconformal and rough geometry Cell decompositions Expansion Thurston maps with two or three postcritical points Visual metrics Symbolic dynamics Tile graphs Isotopies Subdivisions Quotients of Thurston maps Combinatorially expanding Thurston maps Invariant curves The combinatorial expansion factor The measure of maximal entropy The geometry of the visual sphere Rational Thurston maps and Lebesgue measure A combinatorial characterization of Lattes maps Outlook and open problems Appendix A Bibliography Index.

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Expanding Thurston maps

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