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Linearization Models for Complex Dynamical Systems - Topics in Univalent Functions, Functional Equations and Semigroup Theory
Title Page
3
Copyright Page
4
Table of Contents
5
Preface
8
Chapter 1 Geometric Background
12
1.1 Some classes of univalent functions
12
1.1.1 Starlike functions
12
1.1.2 Class S*[0]. Nevanlinna’s condition
13
1.1.3 Classes S*[t ], t . .. Hummel’s representation
14
1.1.4 Spirallike functions. Spa cek’s condition
15
1.1.5 Close-to-convex and .-like functions
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1.2 Boundary behavior of holomorphic functions
18
1.3 The Julia–Wolff–Carath´eodory and Denjoy–Wolff Theorems
21
1.4 Functions of positive real part
24
Chapter 2 Dynamic Approach
27
2.1 Semigroups and generators
27
2.2 Flow invariance conditions and parametric representations of semigroup generators
29
2.3 The Denjoy–Wolff and Julia–Wolff–Carath´eodory Theorems for semigroups
33
2.4 Generators with boundary null points
35
2.5 Univalent functions and semi-complete vector fields
44
Chapter 3 Starlike Functions with Respect to a Boundary Point
48
3.1 Robertson’s classes. Robertson’s conjecture
48
3.2 Auxiliary lemmas
50
3.3 A generalization of Robertson’s conjecture
53
3.4 Angle distortion theorems
55
3.4.1 Smallest exterior wedge
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3.4.2 Biggest interior wedge
58
3.5 Functions convex in one direction
65
Chapter 4 Spirallike Functions with Respect to a Boundary Point
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4.1 Spirallike domains with respect to a boundary point
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4.2 A characterization of spirallike functions with respect to a boundary point
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4.3 Subordination criteria for the class Spiralµ[1]
81
4.4 Distortion Theorems
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4.4.1 ‘Spiral angle’ distortion theorems
83
4.4.2 Growth estimates for semigroup generators
87
4.4.3 Growth estimates for spirallike functions
89
4.4.4 Classes G(µ, ß)
92
4.5 Covering theorems for starlike and spirallike functions
98
Chapter 5 Koenigs Type Starlike and Spirallike Functions
102
5.1 Schr¨oder’s and Abel’s equations
102
5.2 Remarks on stochastic branching processes
106
5.3 Koenigs’ linearization model for dilation type semigroups. Embeddings
110
5.4 Valiron’s type linearization models for hyperbolic type semigroups. Embeddings
112
5.5 Pommerenke’s and Baker–Pommerenke’s linearization models for semigroups with a boundary sink point
119
5.5.1 Pommerenke’s linearization model for automorphic type mappings
119
5.5.2 Baker–Pommerenke’s model for non-automorphic type self-mappings
123
5.5.3 Higher order angular differentiability at boundary fixed points. A unified model
124
5.6 Embedding property via Abel’s equation
126
Chapter 6 Rigidity of Holomorphic Mappings and Commuting Semigroups
128
6.1 The Burns–Krantz theorem
129
6.2 Rigidity of semigroup generators
135
6.3 Commuting semigroups of holomorphic mappings
140
6.3.1 Identity principles for commuting semigroups
140
6.3.2 Dilation type
147
6.3.3 Hyperbolic type
151
6.3.4 Parabolic type
153
Chapter 7 Asymptotic Behavior of One-parameter Semigroups
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7.1 Dilation case
160
7.1.1 General remarks and rates of convergence
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7.1.2 Argument rigidity principle
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7.2 Hyperbolic case
165
7.2.1 Criteria for the exponential convergence
165
7.2.2 Angular similarity principle
174
7.3 Parabolic case
179
7.3.1 Discrete case
179
7.3.2 Continuous case
182
7.3.3 Universal asymptotes
190
Chapter 8 Backward Flow Invariant Domains for Semigroups
201
8.1 Existence
201
8.2 Maximal FIDs. Flower structures
211
8.3 Examples
214
8.4 Angular characteristics of flow invariant domains
217
8.5 Additional remarks
222
Chapter 9 Appendices
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9.1 Controlled Approximation Problems
226
9.1.1 Setting of approximation problems
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9.1.2 Solutions of approximation problems
228
9.1.3 Perturbation formulas
236
9.2 Weighted semigroups of composition operators
245
Bibliography
252
Subject Index
262
Author Index
266
Symbols
268
List of Figures
270
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