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Expanding Thurston Maps Mathematical Surveys and Monographs Volume 225 Expanding Thurston Maps Mario Bonk Daniel Meyer American Mathematical Society 10.1090/surv/225 Expanding Thurston Maps Mario Bonk Daniel Meyer Mathematical Surveys and Monographs Volume 225 Expanding Thurston Maps Mario Bonk Daniel Meyer American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Robert Guralnick Benjamin Sudakov Michael A. Singer, Chair Constantin Teleman MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 37-02, 37F10, 37F20, 30D05, 30L10. For additional information and updates on this book, visit www.ams.org/bookpages/surv-225 Library of Congress Cataloging-in-Publication Data Names: Bonk, Mario. | Meyer, Daniel, 1969– Title: Expanding Thurston maps / Mario Bonk, Daniel Meyer. Description: Providence, Rhode Island: American Mathematical Society, [2017] | Series: Mathe- matical surveys and monographs; volume 225 | Includes bibliographical references and index. Identifiers: LCCN 2017017476 | ISBN 9780821875544 (alk. paper) Subjects: LCSH: Algebraic topology. | Mappings (Mathematics) | AMS: Dynamical systems and ergodic theory – Research exposition (monographs, survey articles). msc | Dynamical systems and ergodic theory – Complex dynamical systems – Polynomials; rational maps; entire and meromorphic functions. msc | Dynamical systems and ergodic theory – Complex dynamical systems – Combinatorics and topology. msc | Functions of a complex variable – Entire and meromorphic functions, and related topics – Functional equations in the complex domain, iteration and composition of analytic functions. msc | Functions of a complex variable – Analysis on metric spaces – Quasiconformal mappings in metric spaces. msc Classification: LCC QA612 .B66 2017 | DDC 515/.39–dc23 LC record available at https://lccn.loc.gov/2017017476 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2017 by the authors. All rights reserved. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 222120191817 Contents List of Figures ix Preface xi Notation xiii Chapter 1. Introduction 1 1.1. A Latt`es map as a first example 3 1.2. Cell decompositions 6 1.3. Fractal spheres 7 1.4. Visual metrics and the visual sphere 11 1.5. Invariant curves 15 1.6. Miscellaneous results 17 1.7. Characterizations of Latt`es maps 19 1.8. Outline of the presentation 21 1.9. List of examples for Thurston maps 26 Chapter 2. Thurston maps 29 2.1. Branched covering maps 29 2.2. Definition of Thurston maps 30 2.3. Definition of expansion 32 2.4. Thurston equivalence 34 2.5. The orbifold associated with a Thurston map 39 2.6. Thurston’s characterization of rational maps 45 Chapter 3. Latt`es maps 49 3.1. Crystallographic groups and Latt`es maps 53 3.2. Quotients of torus endomorphisms and parabolicity 60 3.3. Classifying Latt`es maps 65 3.4. Latt`es-type maps 69 3.5. Covers of parabolic orbifolds 77 3.6. Examples of Latt`es maps 83 Chapter 4. Quasiconformal and rough geometry 89 4.1. Quasiconformal geometry 89 4.2. Gromov hyperbolicity 94 4.3. Gromov hyperbolic groups and Cannon’s conjecture 96 4.4. Quasispheres 98 Chapter 5. Cell decompositions 103 5.1. Cell decompositions in general 104 v vi CONTENTS 5.2. Cell decompositions of 2-spheres 107 5.3. Cell decompositions induced by Thurston maps 114 5.4. Labelings 124 5.5. Thurston maps from cell decompositions 130 5.6. Flowers 135 5.7. Joining opposite sides 139 Chapter 6. Expansion 143 6.1. Definition of expansion revisited 143 6.2. Further results on expansion 147 6.3. Latt`es-type maps and expansion 152 Chapter 7. Thurston maps with two or three postcritical points 159 7.1. Thurston equivalence to rational maps 160 7.2. Thurston maps with signature (∞, ∞)or(2, 2, ∞) 161 Chapter 8. Visual Metrics 169 8.1. The number m(x, y) 172 8.2. Existence and basic properties of visual metrics 175 8.3. The canonical orbifold metric as a visual metric 180 Chapter 9. Symbolic dynamics 185 Chapter 10. Tile graphs 191 Chapter 11. Isotopies 199 11.1. Equivalent expanding Thurston maps are conjugate 200 11.2. Isotopies of Jordan curves 205 11.3. Isotopies and cell decompositions 209 Chapter 12. Subdivisions 217 12.1. Thurston maps with invariant curves 220 12.2. Two-tile subdivision rules 229 12.3. Examples of two-tile subdivision rules 240 Chapter 13. Quotients of Thurston maps 251 13.1. Closed equivalence relations and Moore’s theorem 253 13.2. Branched covering maps and continua 256 13.3. Strongly invariant equivalence relations 260 Chapter 14. Combinatorially expanding Thurston maps 267 Chapter 15. Invariant curves 287 15.1. Existence and uniqueness of invariant curves 291 15.2. Iterative construction of invariant curves 300 15.3. Invariant curves are quasicircles 309 Chapter 16. The combinatorial expansion factor 315 Chapter 17. The measure of maximal entropy 327 17.1. Review of measure-theoretic dynamics 328 17.2. Construction of the measure of maximal entropy 331 CONTENTS vii Chapter 18. The geometry of the visual sphere 345 18.1. Linear local connectedness 347 18.2. Doubling and Ahlfors regularity 350 18.3. Quasisymmetry and rational Thurston maps 352 Chapter 19. Rational Thurston maps and Lebesgue measure 361 19.1. The Jacobian of a measurable map 362 19.2. Ergodicity of Lebesgue measure 364 19.3. The absolutely continuous invariant measure 367 19.4. Latt`es maps, entropy, and Lebesgue measure 377 Chapter 20. A combinatorial characterization of Latt`es maps 385 20.1. Visual metrics, 2-regularity, and Latt`es maps 386 20.2. Separating sets with tiles 389 20.3. Short e-chains 396 Chapter 21. Outlook and open problems 401 Appendix A 413 A.1. Conformal metrics 413 A.2. Koebe’s distortion theorem 415 A.3. Janiszewski’s lemma 418 A.4. Orientations on surfaces 420 A.5. Covering maps 424 A.6. Branched covering maps 425 A.7. Quotient spaces and group actions 439 A.8. Lattices and tori 443 A.9. Orbifolds and coverings 447 A.10. The canonical orbifold metric 453 Bibliography 467 Index 473 List of Figures 1.1 The Latt`es map g.3 1.2 The map h.8 1.3 Polyhedral surfaces obtained from the replacement rule. 9 2.1 The map g.37 2.2 An obstructed map. 47 3.1 Invariant tiling for type (244). 55 3.2 Invariant tiling for type (333). 55 3.3 Invariant tiling for type (236). 55 3.4 Folding a tetrahedron from a triangle. 81 3.5 Construction of Θ = ℘.81 3.6 A Latt`es map with orbifold signature (2, 4, 4). 84 3.7 A Latt`es map with orbifold signature (3, 3, 3). 85 3.8 A Latt`es map with orbifold signature (2, 3, 6). 85 3.9 A Euclidean model for a flexible Latt`es map. 87 3.10 The map f in Example 3.27. 88 4.1 The generator of the snowsphere S.99 4.2 The set Z. 101 5.1 The cycle of a vertex v. 110 5.2 A chain, an n-chain, and an e-chain. 123 6.1 A map with a Levy cycle. 151 7.1 Model for a Chebyshev polynomial. 166 8.1 Separating points by tiles. 170 11.1 Tower of isotopies. 203 11.2 J is not isotopic to S1 rel. {1, i, −1, −i}. 206 11.3 Constructing a path through a, b, p. 210 11.4 Construction of the curve C. 214 12.1 Subdividing tiles. 223 ix x LIST OF FIGURES 12.2 The proof of Lemma 12.8. 226 12.3 Two subdivision rules. 230 12.4 The two-tile subdivision rule for z2 − 1. 241 12.5 Tiles of level 7 for Example 12.20. 242 12.6 The barycentric subdivision rule. 243 12.7 Tiles of levels 1–6 for the barycentric subdivision map f2. 244 12.8 The 2-by-3 subdivision rule. 245 12.9 Adding flaps. 247 12.10 Two-tile subdivision rule realized by g7. 248 14.1 Equivalence classes of vertex-, edge-, and tile-type. 272 14.2 A two-tile subdivision rule realized by a map g with post(g) = V0. 283 14.3 The map f is not combinatorially expanding, but equivalent to the expanding map g. 284 15.1 The invariant curve for Example 15.6. 290 15.2 Invariant curves for the Latt`es map g. 293 15.3 No invariant Jordan curve C⊃ post(f). 295 15.4 Iterative construction of an invariant curve. 301 15.5 Iterative construction by replacing edges. 306 15.6 Example where C is not a Jordan curve. 308 15.7 A non-trivial rectifiable invariant Jordan curve.
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