Linearization Models for Complex Dynamical Systems

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ISBN: 9783034605083 | 3034605080
Cover: Hardcover
Copyright: 6/28/2010

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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.

Preface	p. ix
Geometric Background	p. 1
Some classes of univalent functions	p. 1
Starlike functions	p. 1
Class S*[0]. Nevanlinna's condition	p. 2
Classes S*[¿], ¿ ¿ ¿. Hummel's representation	p. 3
Spirallike functions. ¿pacek's condition	p. 4
Close-to-convex and ¿-like functions	p. 6
Boundary behavior of holomorphic functions	p. 7
The Julia-Wolff-Carathéodory and Denjoy-Wolff Theorems	p. 10
Functions of positive real part	p. 13
Dynamic Approach	p. 17
Semigroups and generators	p. 17
Flow invariance conditions and parametric representations of semigroup generators	p. 19
The Denjoy-Wolff and Julia-Wolff-Carathéodory Theorems for semigroups	p. 23
Generators with boundary null points	p. 25
Univalent functions and semi-complete vector fields	p. 34
Starlike Functions with Respect to a Boundary Point	p. 39
Robertson's classes. Robertson's conjecture	p. 39
Auxiliary lemmas	p. 41
A generalization of Robertson's conjecture	p. 44
Angle distortion theorems	p. 46
Smallest exterior wedge	p. 46
Biggest interior wedge	p. 49
Functions convex in one direction	p. 56
Spirallike Functions with Respect to a Boundary Point	p. 63
Spirallike domains with respect to a boundary point	p. 63
A characterization of spirallike functions with respect to a boundary point	p. 69
Subordination criteria for the class Spiral_¿[1]	p. 73
Distortion Theorems	p. 75
'Spiral angle' distortion theorems	p. 75
Growth estimates for semigroup generators	p. 79
Growth estimates for spirallike functions	p. 81
Classes G(¿, ß)	p. 84
Covering theorems for starlike and spirallike functions	p. 90
K¿nigs Type Starlike and Spirallike Functions	p. 95
Schröder's and Abel's equations	p. 95
Remarks on stochastic branching processes	p. 99
K¿nigs' linearization model for dilation type semigroups. Embeddings	p. 103
Valiron's type linearization models for hyperbolic type semigroups. Embeddings	p. 105
Pommerenke's and Baker-Pommerenke's linearization models for semigroups with a boundary sink point	p. 112
Pommerenke's linearization model for automorphic type mappings	p. 112
Baker-Pommerenke's model for non-automorphic type self-mappings	p. 116
Higher order angular differentiability at boundary fixed points. A unified model	p. 117
Embedding property via Abel's equation	p. 119
Rigidity of Holomorphic Mappings and Commuting Semigroups	p. 121
The Burns-Krantz theorem	p. 122
Rigidity of semigroup generators	p. 128
Commuting semigroups of holomorphic mappings	p. 133
Identity principles for commuting semigroups	p. 133
Dilation type	p. 140
Hyperbolic type	p. 144
Parabolic type	p. 146
Asymptotic Behavior of One-parameter Semigroups	p. 153
Dilation case	p. 154
General remarks and rates of convergence	p. 154
Argument rigidity principle	p. 157
Hyperbolic case	p. 159
Criteria for the exponential convergence	p. 159
Angular similarity principle	p. 168
Parabolic case	p. 173
Discrete case	p. 173
Continuous case	p. 176
Universal asymptotes	p. 184
Backward Flow Invariant Domains for Semigroups	p. 195
Existence	p. 195
Maximal FIDs. Flower structures	p. 205
Examples	p. 208
Angular characteristics of flow invariant domains	p. 211
Additional remarks	p. 216
Appendices	p. 221
Controlled Approximation Problems	p. 221
Setting of approximation Problems	p. 221
Solutions of approximation problems	p. 223
Perturbation formulas	p. 231
Weighted semigroups of composition operators	p. 240
Bibliography	p. 247
Subject Index	p. 257
Author Index	p. 261
Symbols	p. 263
List of Figures	p. 265
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Linearization Models for Complex Dynamical Systems

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