Linearization Models for Complex Dynamical Systems - Topics in Univalent Functions, Functional Equations and Semigroup Theory

Title Page

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Table of Contents

Preface

Chapter 1 Geometric Background

1.1 Some classes of univalent functions

1.1.1 Starlike functions

1.1.2 Class S*[0]. Nevanlinna’s condition

1.1.3 Classes S*[t ], t . .. Hummel’s representation

1.1.4 Spirallike functions. Spa cek’s condition

1.1.5 Close-to-convex and .-like functions

1.2 Boundary behavior of holomorphic functions

1.3 The Julia–Wolff–Carath´eodory and Denjoy–Wolff Theorems

1.4 Functions of positive real part

Chapter 2 Dynamic Approach

2.1 Semigroups and generators

2.2 Flow invariance conditions and parametric representations of semigroup generators

2.3 The Denjoy–Wolff and Julia–Wolff–Carath´eodory Theorems for semigroups

2.4 Generators with boundary null points

2.5 Univalent functions and semi-complete vector fields

Chapter 3 Starlike Functions with Respect to a Boundary Point

3.1 Robertson’s classes. Robertson’s conjecture

3.2 Auxiliary lemmas

3.3 A generalization of Robertson’s conjecture

3.4 Angle distortion theorems

3.4.1 Smallest exterior wedge

3.4.2 Biggest interior wedge

3.5 Functions convex in one direction

Chapter 4 Spirallike Functions with Respect to a Boundary Point

4.1 Spirallike domains with respect to a boundary point

4.2 A characterization of spirallike functions with respect to a boundary point

4.3 Subordination criteria for the class Spiralµ[1]

4.4 Distortion Theorems

4.4.1 ‘Spiral angle’ distortion theorems

4.4.2 Growth estimates for semigroup generators

4.4.3 Growth estimates for spirallike functions

4.4.4 Classes G(µ, ß)

4.5 Covering theorems for starlike and spirallike functions

Chapter 5 Koenigs Type Starlike and Spirallike Functions

5.1 Schr¨oder’s and Abel’s equations

5.2 Remarks on stochastic branching processes

5.3 Koenigs’ linearization model for dilation type semigroups. Embeddings

5.4 Valiron’s type linearization models for hyperbolic type semigroups. Embeddings

5.5 Pommerenke’s and Baker–Pommerenke’s linearization models for semigroups with a boundary sink point

5.5.1 Pommerenke’s linearization model for automorphic type mappings

5.5.2 Baker–Pommerenke’s model for non-automorphic type self-mappings

5.5.3 Higher order angular differentiability at boundary fixed points. A unified model

5.6 Embedding property via Abel’s equation

Chapter 6 Rigidity of Holomorphic Mappings and Commuting Semigroups

6.1 The Burns–Krantz theorem

6.2 Rigidity of semigroup generators

6.3 Commuting semigroups of holomorphic mappings

6.3.1 Identity principles for commuting semigroups

6.3.2 Dilation type

6.3.3 Hyperbolic type

6.3.4 Parabolic type

Chapter 7 Asymptotic Behavior of One-parameter Semigroups

7.1 Dilation case

7.1.1 General remarks and rates of convergence

7.1.2 Argument rigidity principle

7.2 Hyperbolic case

7.2.1 Criteria for the exponential convergence

7.2.2 Angular similarity principle

7.3 Parabolic case

7.3.1 Discrete case

7.3.2 Continuous case

7.3.3 Universal asymptotes

Chapter 8 Backward Flow Invariant Domains for Semigroups

8.1 Existence

8.2 Maximal FIDs. Flower structures

8.3 Examples

8.4 Angular characteristics of flow invariant domains

8.5 Additional remarks

Chapter 9 Appendices

9.1 Controlled Approximation Problems

9.1.1 Setting of approximation problems

9.1.2 Solutions of approximation problems

9.1.3 Perturbation formulas

9.2 Weighted semigroups of composition operators

Bibliography

Subject Index

Author Index

Symbols

List of Figures