**Contemporary Mathematics**

Volume: 573;
2012;
256 pp;
Softcover

MSC: Primary 30; 32; 37;

Print ISBN: 978-0-8218-5348-1

Product Code: CONM/573

List Price: $86.00

Individual Member Price: $68.80

**Electronic ISBN: 978-0-8218-9025-7
Product Code: CONM/573.E**

List Price: $86.00

Individual Member Price: $68.80

# Conformal Dynamics and Hyperbolic Geometry

Share this page *Edited by *
*Francis Bonahon; Robert L. Devaney; Frederick P. Gardiner; Dragomir Šarić*

This volume contains the proceedings of the
Conference on Conformal Dynamics and Hyperbolic Geometry, held October
21–23, 2010, in honor of Linda Keen's 70th birthday.

This volume provides a valuable introduction to problems in
conformal and hyperbolic geometry and one dimensional, conformal
dynamics. It includes a classic expository article by John Milnor on
the structure of hyperbolic components of the parameter space for
dynamical systems arising from the iteration of polynomial maps in the
complex plane. In addition there are foundational results concerning
Teichmüller theory, the geometry of Fuchsian and Kleinian groups,
domain convergence properties for the Poincaré metric, elaboration
of the theory of the universal solenoid, the geometry of dynamical
systems acting on a circle, and realization of Thompson's group as a
mapping class group for a uniformly asymptotically affine circle
endomorphism.

The portion of the volume dealing with complex dynamics will appeal
to a diverse group of mathematicians. Recently many researchers
working in a wide range of topics, including topology, algebraic
geometry, complex analysis, and dynamical systems, have become involved
in aspects of this field.

#### Table of Contents

# Table of Contents

## Conformal Dynamics and Hyperbolic Geometry

- Preface vii8 free
- Normal families and holomorphic motions over infinite dimensional parameter spaces 110 free
- Elementary moves and the modular group of the compact solenoid 1120
- 1. Introduction 1120
- 2. The abstract commensurator 𝐶𝑜𝑚𝑚(𝜋₁(𝑆)) of a surface group 1423
- 3. Pants decompositions and subgroups of surface groups 1524
- 4. Elementary moves and induced isomorphisms of finite index subgroups of 𝜋₁(𝑆) 2029
- 5. The isotropy group 2231
- 6. Exact sequences 2837
- 7. The center of 𝑀𝑜𝑑(ℋ) is trivial 3039
- 8. Relations in 𝑀𝑜𝑑(ℋ) 3140
- References 3241

- Combinatorics and topology of the shift locus 3544
- Dynamics of 𝑧ⁿ+𝜆/𝑧ⁿ; Why the Case 𝑛=2 is Crazy 4958
- On holomorphic families of Riemann surfaces 6776
- Circle Endomorphisms, Dual Circles and Thompson’s Group 99108
- Introduction 99108
- 1. Circle endomorphisms 101110
- 2. Rohlin formula for smooth expanding circle endomorphisms 103112
- 3. The symbolic dynamical system 106115
- 4. Dual dynamical systems and scaling functions 107116
- 5. Scaling functions on the dual circle 109118
- 6. Teichmüller’s metric on 𝑈𝐴𝐴. 110119
- 7. The action of Thompson’s 𝐹-group. 110119
- 8. The dual Rohlin formula 113122
- References 116125

- Rational maps with half symmetries, Julia sets, and Multibrot sets in parameter planes 119128
- The rate of convergence of the hyperbolic density on sequences of domains. 147156
- The asymptotic directions of pleating rays in the Maskit embedding. 159168
- Hyperbolic components 183192
- 1. Introduction. 183192
- 2. The Affine Parameter Spaces 𝒫^{𝒹} and 𝒫^{𝒮₀}. 185194
- 3. Graphs and Symmetries. 187196
- 4. Blaschke Products and the Model Space ℬ^{𝒮}. 190199
- 5. Hyperbolic Components are Topological Cells. 196205
- 6. Analytic Isomorphism between Hyperbolic Components. 202211
- 7. Real Forms. 205214
- 8. Polynomials with Marked Critical Points. 214223
- 9. Rational Maps. 215224
- Appendix A. Realizing Reduced Schemes (by Alfredo Poirier) 223232
- Appendix B. Census of Reduced Schemes 227236
- References 231240

- On barycenter entropy for rational maps 233242
- Parameter Plane of a Family of Meromorphic Functions with Two Asymptotic Values 245254

#### Readership

Graduate students and research mathematicians interested in conformal dynamics and hyperbolic geometry.