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Hardcover ISBN:  9780821875544 
Product Code:  SURV/225 
List Price:  $169.00 
MAA Member Price:  $152.10 
AMS Member Price:  $135.20 
eBook ISBN:  9781470442521 
Product Code:  SURV/225.E 
List Price:  $159.00 
MAA Member Price:  $143.10 
AMS Member Price:  $127.20 
Hardcover ISBN:  9780821875544 
eBook ISBN:  9781470442521 
Product Code:  SURV/225.B 
List Price:  $328.00 $248.50 
MAA Member Price:  $295.20 $223.65 
AMS Member Price:  $262.40 $198.80 

Book DetailsMathematical Surveys and MonographsVolume: 225; 2017; 478 ppMSC: Primary 37; 30
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 twodimensional topological sphere such that each critical point of the map has a finite orbit under iteration. It 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.
The book is an introduction to the subject. The prerequisites for the reader are modest and include some basic knowledge of complex analysis and topology. The book has an extensive appendix, where background material is reviewed such as orbifolds and branched covering maps.
ReadershipGraduate students and researchers interested in dynamical systems and topological dynamics.

Table of Contents

Chapters

Introduction

Thurston maps

Lattès 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 Lattès maps

Outlook and open problems

Appendix A


Additional Material

Reviews

A novice reader working through this text will find a gentle introduction to these topics in a concrete intuitive setting.
Kevin M. Pilgrim, Mathematical Reviews


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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 twodimensional topological sphere such that each critical point of the map has a finite orbit under iteration. It 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.
The book is an introduction to the subject. The prerequisites for the reader are modest and include some basic knowledge of complex analysis and topology. The book has an extensive appendix, where background material is reviewed such as orbifolds and branched covering maps.
Graduate students and researchers interested in dynamical systems and topological dynamics.

Chapters

Introduction

Thurston maps

Lattès 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 Lattès maps

Outlook and open problems

Appendix A

A novice reader working through this text will find a gentle introduction to these topics in a concrete intuitive setting.
Kevin M. Pilgrim, Mathematical Reviews