**Mathematical Surveys and Monographs**

Volume: 225;
2017;
478 pp;
Hardcover

MSC: Primary 37; 30;

**Print ISBN: 978-0-8218-7554-4
Product Code: SURV/225**

List Price: $169.00

AMS Member Price: $135.20

MAA Member Price: $152.10

**Electronic ISBN: 978-1-4704-4252-1
Product Code: SURV/225.E**

List Price: $169.00

AMS Member Price: $135.20

MAA Member Price: $152.10

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#### Supplemental Materials

# Expanding Thurston Maps

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*Mario Bonk; Daniel Meyer*

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. 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.

#### Readership

Graduate students and researchers interested in dynamical systems and topological dynamics.

#### Reviews & Endorsements

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

#### Table of Contents

# Table of Contents

## Expanding Thurston Maps

- Cover Cover11
- Title page iii4
- Contents v6
- List of Figures ix10
- Preface xi12
- Notation xiii14
- Chapter 1. Introduction 118
- 1.1. A Lattès map as a first example 320
- 1.2. Cell decompositions 623
- 1.3. Fractal spheres 724
- 1.4. Visual metrics and the visual sphere 1128
- 1.5. Invariant curves 1532
- 1.6. Miscellaneous results 1734
- 1.7. Characterizations of Lattès maps 1936
- 1.8. Outline of the presentation 2138
- 1.9. List of examples for Thurston maps 2643

- Chapter 2. Thurston maps 2946
- Chapter 3. Lattès maps 4966
- Chapter 4. Quasiconformal and rough geometry 89106
- Chapter 5. Cell decompositions 103120
- Chapter 6. Expansion 143160
- Chapter 7. Thurston maps with two or three postcritical points 159176
- Chapter 8. Visual Metrics 169186
- Chapter 9. Symbolic dynamics 185202
- Chapter 10. Tile graphs 191208
- Chapter 11. Isotopies 199216
- Chapter 12. Subdivisions 217234
- Chapter 13. Quotients of Thurston maps 251268
- Chapter 14. Combinatorially expanding Thurston maps 267284
- Chapter 15. Invariant curves 287304
- Chapter 16. The combinatorial expansion factor 315332
- Chapter 17. The measure of maximal entropy 327344
- Chapter 18. The geometry of the visual sphere 345362
- Chapter 19. Rational Thurston maps and Lebesgue measure 361378
- Chapter 20. A combinatorial characterization of Lattès maps 385402
- Chapter 21. Outlook and open problems 401418
- Appendix A 413430
- A.1. Conformal metrics 413430
- A.2. Koebe’s distortion theorem 415432
- A.3. Janiszewski’s lemma 418435
- A.4. Orientations on surfaces 420437
- A.5. Covering maps 424441
- A.6. Branched covering maps 425442
- A.7. Quotient spaces and group actions 439456
- A.8. Lattices and tori 443460
- A.9. Orbifolds and coverings 447464
- A.10. The canonical orbifold metric 453470

- Bibliography 467484
- Index 473490
- Back Cover Back Cover1497