Holomorphic Dynamics on Hyperbolic Riemann Surfaces

✍ Scribed by Marco Abate

• Publisher: De Gruyter
• Year: 2022
• Tongue: English
• Leaves: 371
• Series: de Gruyter Studies in Mathematics 89
• Category: Library

✦ Synopsis

This completely revised and updated edition of the one variable part of the author's classic older book "Iteration Theory of Holomorphic Maps on Taut Manifolds" presents the theory of holomorphic dynamical systems on hyperbolic Riemann surfaces from the very beginning of the subject up to the most recent developments. It is intended both as a reference book for the experts and as an accessible gateway to this beautiful theory for Master and Ph.D. students. It also contains extensive historical notes and references for further readings.

✦ Table of Contents

Contents
Introduction
1. The Schwarz lemma and Riemann surfaces
1.1 The Schwarz–Pick lemma
1.2 The Poincaré distance
1.3 The upper half-plane
1.4 Fixed points of automorphisms
1.5 Multipoint Schwarz–Pick lemmas
1.6 Riemann surfaces
1.7 Hyperbolic Riemann surfaces and the Montel theorem
1.8 Boundary behavior of the universal covering map
1.9 The Poincaré metric
1.10 The Ahlfors lemma
1.11 Bloch domains
2. Boundary Schwarz lemmas
2.1 The Julia lemma
2.2 Stolz regions and nontangential limits
2.3 The Julia–Wolff–Carathéodory theorem
2.4 The Lindelöf theorem
2.5 The Wolff lemma
2.6 The automorphism group of hyperbolic Riemann surfaces
2.7 The Burns–Krantz theorem
3. Discrete dynamics on Riemann surfaces
3.1 The fixed-point case
3.2 The Wolff–Denjoy theorem
3.3 The Heins theorem
3.4 Stability of the Wolff point
3.5 Models on Riemann surfaces
3.6 Random iteration on Bloch domains
3.7 Random iteration of small perturbations
4. Discrete dynamics on the unit disk
4.1 Elliptic dynamics
4.2 Superattracting dynamics
4.3 Hyperbolic dynamics
4.4 Parabolic dynamics
4.5 Models on the unit disk
4.6 The hyperbolic step
4.7 Parabolic type and boundary smoothness
4.8 Boundary fixed points
4.9 Backward dynamics
4.10 Commuting functions
5. Continuous dynamics on Riemann surfaces
5.1 Algebraic semigroup homomorphisms
5.2 One-parameter semigroups
5.3 One-parameter semigroups on Riemann surfaces
5.4 The infinitesimal generator
5.5 The continuous Wolff–Denjoy theorem
5.6 The Berkson–Porta formula
5.7 One-parameter semigroups on the unit disk
A. Appendix
A.1 The Hurwitz theorems
A.2 The Fatou uniqueness theorem
A.3 Holomorphic functions with nonnegative real part
A.4 Sequences
A.5 Topological groups
Bibliography
[27]
[53]
[78]
[105]
[132]
[163]
[188]
[214]
[242]
[269]
[297]
[323]
[351]
[380]
[409]
Index
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