Elin / Shoikhet Linearization Models for Complex Dynamical Systems

Topics in Univalent Functions, Functional Equations and Semigroup Theory
2010
ISBN: 978-3-0346-0509-0
Verlag: Birkhäuser Basel
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 208, 268 Seiten

Reihe: Operator Theory: Advances and Applications

Linearization models for discrete and continuous time dynamical systems are the driving forces for modern geometric function theory and composition operator theory on function spaces. This book focuses on a systematic survey and detailed treatment of linearization models for one-parameter semigroups, Schröder's and Abel's functional equations, and various classes of univalent functions which serve as intertwining mappings for nonlinear and linear semigroups. These topics are applicable to the study of problems in complex analysis, stochastic and evolution processes and approximation theory.

Autoren/Hrsg.

Elin, Mark

Shoikhet, David

Inhaltsverzeichnis

1;Title Page ;3
2;Copyright Page ;4
3;Table of Contents ;5
4;Preface;8
5;Chapter 1 Geometric Background;12
5.1;1.1 Some classes of univalent functions;12
5.1.1;1.1.1 Starlike functions;12
5.1.2;1.1.2 Class S*[0]. Nevanlinna’s condition;13
5.1.3;1.1.3 Classes S*[t ], t . .. Hummel’s representation;14
5.1.4;1.1.4 Spirallike functions. Spa cek’s condition;15
5.1.5;1.1.5 Close-to-convex and .-like functions;17
5.2;1.2 Boundary behavior of holomorphic functions;18
5.3;1.3 The Julia–Wolff–Carath´eodory and Denjoy–Wolff Theorems;21
5.4;1.4 Functions of positive real part;24
6;Chapter 2 Dynamic Approach;27
6.1;2.1 Semigroups and generators;27
6.2;2.2 Flow invariance conditions and parametric representations of semigroup generators;29
6.3;2.3 The Denjoy–Wolff and Julia–Wolff–Carath´eodory Theorems for semigroups;33
6.4;2.4 Generators with boundary null points;35
6.5;2.5 Univalent functions and semi-complete vector fields;44
7;Chapter 3 Starlike Functions with Respect to a Boundary Point;48
7.1;3.1 Robertson’s classes. Robertson’s conjecture;48
7.2;3.2 Auxiliary lemmas;50
7.3;3.3 A generalization of Robertson’s conjecture;53
7.4;3.4 Angle distortion theorems;55
7.4.1;3.4.1 Smallest exterior wedge;55
7.4.2;3.4.2 Biggest interior wedge;58
7.5;3.5 Functions convex in one direction;65
8;Chapter 4 Spirallike Functions with Respect to a Boundary Point;71
8.1;4.1 Spirallike domains with respect to a boundary point;71
8.2;4.2 A characterization of spirallike functions with respect to a boundary point;77
8.3;4.3 Subordination criteria for the class Spiralµ[1];81
8.4;4.4 Distortion Theorems;83
8.4.1;4.4.1 ‘Spiral angle’ distortion theorems;83
8.4.2;4.4.2 Growth estimates for semigroup generators;87
8.4.3;4.4.3 Growth estimates for spirallike functions;89
8.4.4;4.4.4 Classes G(µ, ß);92
8.4.5;4.5 Covering theorems for starlike and spirallike functions ;98
9;Chapter 5 Koenigs Type Starlike and Spirallike Functions;102
9.1;5.1 Schr¨oder’s and Abel’s equations;102
9.2;5.2 Remarks on stochastic branching processes;106
9.3;5.3 Koenigs’ linearization model for dilation type semigroups. Embeddings;110
9.4;5.4 Valiron’s type linearization models for hyperbolic type semigroups. Embeddings;112
9.5;5.5 Pommerenke’s and Baker–Pommerenke’s linearization models for semigroups with a boundary sink point;119
9.5.1;5.5.1 Pommerenke’s linearization model for automorphic type mappings;119
9.5.2;5.5.2 Baker–Pommerenke’s model for non-automorphic type self-mappings;123
9.5.3;5.5.3 Higher order angular differentiability at boundary fixed points. A unified model;124
9.6;5.6 Embedding property via Abel’s equation;126
10;Chapter 6 Rigidity of Holomorphic Mappings and Commuting Semigroups;128
10.1;6.1 The Burns–Krantz theorem;129
10.2;6.2 Rigidity of semigroup generators;135
10.3;6.3 Commuting semigroups of holomorphic mappings;140
10.3.1;6.3.1 Identity principles for commuting semigroups;140
10.3.2;6.3.2 Dilation type;147
10.3.3;6.3.3 Hyperbolic type;151
10.3.4;6.3.4 Parabolic type;153
11;Chapter 7 Asymptotic Behavior of One-parameter Semigroups;159
11.1;7.1 Dilation case;160
11.1.1;7.1.1 General remarks and rates of convergence;160
11.1.2;7.1.2 Argument rigidity principle;163
11.2;7.2 Hyperbolic case;165
11.2.1;7.2.1 Criteria for the exponential convergence;165
11.2.2;7.2.2 Angular similarity principle;174
11.3;7.3 Parabolic case;179
11.3.1;7.3.1 Discrete case;179
11.3.2;7.3.2 Continuous case;182
11.3.3;7.3.3 Universal asymptotes;190
12;Chapter 8 Backward Flow Invariant Domains for Semigroups;201
12.1;8.1 Existence;201
12.2;8.2 Maximal FIDs. Flower structures;211
12.3;8.3 Examples;214
12.4;8.4 Angular characteristics of flow invariant domains;217
12.5;8.5 Additional remarks;222
13;Chapter 9 Appendices;226
13.1;9.1 Controlled Approximation Problems;226
13.1.1;9.1.1 Setting of approximation problems;226
13.1.2;9.1.2 Solutions of approximation problems;228
13.1.3;9.1.3 Perturbation formulas;236
13.2;9.2 Weighted semigroups of composition operators;245
14;Bibliography;252
15;Subject Index;262
16;Author Index;266
17;Symbols;268
18;List of Figures;270