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E-raamat: Expanding Thurston Maps

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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 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.
List of Figures ix
Preface xi
Notation xiii
Chapter 1 Introduction 1(28)
1.1 A Lattes map as a first example
3(3)
1.2 Cell decompositions
6(1)
1.3 Fractal spheres
7(4)
1.4 Visual metrics and the visual sphere
11(4)
1.5 Invariant curves
15(2)
1.6 Miscellaneous results
17(2)
1.7 Characterizations of Lattes maps
19(2)
1.8 Outline of the presentation
21(5)
1.9 List of examples for Thurston maps
26(3)
Chapter 2 Thurston maps 29(20)
2.1 Branched covering maps
29(1)
2.2 Definition of Thurston maps
30(2)
2.3 Definition of expansion
32(2)
2.4 Thurston equivalence
34(5)
2.5 The orbifold associated with a Thurston map
39(6)
2.6 Thurston's characterization of rational maps
45(4)
Chapter 3 Lattes maps 49(40)
3.1 Crystallographic groups and Lattes maps
53(7)
3.2 Quotients of torus endomorphisms and parabolicity
60(5)
3.3 Classifying Lattes maps
65(4)
3.4 Lattes-type maps
69(8)
3.5 Covers of parabolic orbifolds
77(6)
3.6 Examples of Lattes maps
83(6)
Chapter 4 Quasiconformal and rough geometry 89(14)
4.1 Quasiconformal geometry
89(5)
4.2 Gromov hyperbolicity
94(2)
4.3 Gromov hyperbolic groups and Cannon's conjecture
96(2)
4.4 Quasispheres
98(5)
Chapter 5 Cell decompositions 103(40)
5.1 Cell decompositions in general
104(3)
5.2 Cell decompositions of 2-spheres
107(7)
5.3 Cell decompositions induced by Thurston maps
114(10)
5.4 Labelings
124(6)
5.5 Thurston maps from cell decompositions
130(5)
5.6 Flowers
135(4)
5.7 Joining opposite sides
139(4)
Chapter 6 Expansion 143(16)
6.1 Definition of expansion revisited
143(4)
6.2 Further results on expansion
147(5)
6.3 Lattes-type maps and expansion
152(7)
Chapter 7 Thurston maps with two or three postcritical points 159(10)
7.1 Thurston equivalence to rational maps
160(1)
7.2 Thurston maps with signature (infinity, infinity) or (2, 2, infinity)
161(8)
Chapter 8 Visual Metrics 169(16)
8.1 The number m(x, y)
172(3)
8.2 Existence and basic properties of visual metrics
175(5)
8.3 The canonical orbifold metric as a visual metric
180(5)
Chapter 9 Symbolic dynamics 185(6)
Chapter 10 Tile graphs 191(8)
Chapter 11 Isotopies 199(18)
11.1 Equivalent expanding Thurston maps are conjugate
200(5)
11.2 Isotopies of Jordan curves
205(4)
11.3 Isotopies and cell decompositions
209(8)
Chapter 12 Subdivisions 217(34)
12.1 Thurston maps with invariant curves
220(9)
12.2 Two-tile subdivision rules
229(11)
12.3 Examples of two-tile subdivision rules
240(11)
Chapter 13 Quotients of Thurston maps 251(16)
13.1 Closed equivalence relations and Moore's theorem
253(3)
13.2 Branched covering maps and continua
256(4)
13.3 Strongly invariant equivalence relations
260(7)
Chapter 14 Combinatorially expanding Thurston maps 267(20)
Chapter 15 Invariant curves 287(28)
15.1 Existence and uniqueness of invariant curves
291(9)
15.2 Iterative construction of invariant curves
300(9)
15.3 Invariant curves are quasicircles
309(6)
Chapter 16 The combinatorial expansion factor 315(12)
Chapter 17 The measure of maximal entropy 327(18)
17.1 Review of measure-theoretic dynamics
328(3)
17.2 Construction of the measure of maximal entropy
331(14)
Chapter 18 The geometry of the visual sphere 345(16)
18.1 Linear local connectedness
347(3)
18.2 Doubling and Ahlfors regularity
350(2)
18.3 Quasisymmetry and rational Thurston maps
352(9)
Chapter 19 Rational Thurston maps and Lebesgue measure 361(24)
19.1 The Jacobian of a measurable map
362(2)
19.2 Ergodicity of Lebesgue measure
364(3)
19.3 The absolutely continuous invariant measure
367(10)
19.4 Lattes maps, entropy, and Lebesgue measure
377(8)
Chapter 20 A combinatorial characterization of Lattes maps 385(16)
20.1 Visual metrics, 2-regularity, and Lattes maps
386(3)
20.2 Separating sets with tiles
389(7)
20.3 Short e-chains
396(5)
Chapter 21 Outlook and open problems 401(12)
Appendix A 413(54)
A.1 Conformal metrics
413(2)
A.2 Koebe's distortion theorem
415(3)
A.3 Janiszewski's lemma
418(2)
A.4 Orientations on surfaces
420(4)
A.5 Covering maps
424(1)
A.6 Branched covering maps
425(14)
A.7 Quotient spaces and group actions
439(4)
A.8 Lattices and tori
443(4)
A.9 Orbifolds and coverings
447(6)
A.10 The canonical orbifold metric
453(14)
Bibliography 467(6)
Index 473
Mario Bonk, University of California, Los Angeles, CA.

Daniel Meyer, University of Liverpool, UK.
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