Linearization Models for Complex Dynamical Systems (eBook)

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 17
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 55
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 71
4.1 Spirallike domains with respect to a boundary point 71
4.2 A characterization of spirallike functions with respect to a boundary point 77
4.3 Subordination criteria for the class Spiralµ[1] 81
4.4 Distortion Theorems 83
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 159
7.1 Dilation case 160
7.1.1 General remarks and rates of convergence 160
7.1.2 Argument rigidity principle 163
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 226
9.1 Controlled Approximation Problems 226
9.1.1 Setting of approximation problems 226
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