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

:	Mark Elin, David Shoikhet
:	Linearization Models for Complex Dynamical Systems Topics in Univalent Functions, Functional Equations and Semigroup Theory
:	Birkhäuser Basel
:	9783034605090
:	1
:	CHF 85.40
:

:	Analysis
:	English

:	268
:	Wasserzeichen/DRM
:	PC/MAC/eReader/Tablet
:	PDF

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.

	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