Linearization Models for Complex Dynamical Systems: Topics in Univalent Functions, Functional Equations and Semigroup Theory (Operator Theory: Advances and Applications, 208)

✍ Scribed by Mark Elin, David Shoikhet

• Publisher: Birkhäuser
• Year: 2010
• Tongue: English
• Leaves: 271
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Title Page
Copyright Page
Table of Contents
Preface
Chapter 1 Geometric Background
1.1 Some classes of univalent functions
1.1.1 Starlike functions
1.1.2 Class S∗[0]. Nevanlinna’s condition
1.1.3 Classes S∗[τ ], τ ∈ Δ. Hummel’s representation
1.1.4 Spirallike functions. ˇSpaˇcek’s condition
1.1.5 Close-to-convex and ϕ-like functions
1.2 Boundary behavior of holomorphic functions
1.3 The Julia–Wolff–Carath´eodory and Denjoy–Wolff Theorems
1.4 Functions of positive real part
Chapter 2 Dynamic Approach
2.1 Semigroups and generators
2.2 Flow invariance conditions and parametric representations of semigroup generators
2.3 The Denjoy–Wolff and Julia–Wolff–Carath´eodory Theorems for semigroups
2.4 Generators with boundary null points
2.5 Univalent functions and semi-complete vector fields
Chapter 3 Starlike Functions with Respect to a Boundary Point
3.1 Robertson’s classes. Robertson’s conjecture
3.2 Auxiliary lemmas
3.3 A generalization of Robertson’s conjecture
3.4 Angle distortion theorems
3.4.1 Smallest exterior wedge
3.4.2 Biggest interior wedge
3.5 Functions convex in one direction
Chapter 4 Spirallike Functions with Respect to a Boundary Point
4.1 Spirallike domains with respect to a boundary point
4.2 A characterization of spirallike functions with respect to a boundary point
4.3 Subordination criteria for the class Spiralμ[1]
4.4 Distortion Theorems
4.4.1 ‘Spiral angle’ distortion theorems
4.4.2 Growth estimates for semigroup generators
4.4.3 Growth estimates for spirallike functions
4.4.4 Classes G(μ, β)
4.5 Covering theorems for starlike and spirallike functions
Chapter 5 Koenigs Type Starlike and Spirallike Functions
5.1 Schr¨oder’s and Abel’s equations
5.2 Remarks on stochastic branching processes
5.3 Koenigs’ linearization model for dilation type semigroups. Embeddings
5.4 Valiron’s type linearization models for hyperbolic type semigroups. Embeddings
5.5 Pommerenke’s and Baker–Pommerenke’s linearization models for semigroups with a boundary sink point
5.5.1 Pommerenke’s linearization model for automorphic type mappings
5.5.2 Baker–Pommerenke’s model for non-automorphic type self-mappings
5.5.3 Higher order angular differentiability at boundary fixed points. A unified model
5.6 Embedding property via Abel’s equation
Chapter 6 Rigidity of Holomorphic Mappings and Commuting Semigroups
6.1 The Burns–Krantz theorem
6.2 Rigidity of semigroup generators
6.3 Commuting semigroups of holomorphic mappings
6.3.1 Identity principles for commuting semigroups
6.3.2 Dilation type
6.3.3 Hyperbolic type
6.3.4 Parabolic type
Chapter 7 Asymptotic Behavior of One-parameter Semigroups
7.1 Dilation case
7.1.1 General remarks and rates of convergence
7.1.2 Argument rigidity principle
7.2 Hyperbolic case
7.2.1 Criteria for the exponential convergence
7.2.2 Angular similarity principle
7.3 Parabolic case
7.3.1 Discrete case
7.3.2 Continuous case
7.3.3 Universal asymptotes
Chapter 8 Backward Flow Invariant Domains for Semigroups
8.1 Existence
8.2 Maximal FIDs. Flower structures
8.3 Examples
8.4 Angular characteristics of flow invariant domains
8.5 Additional remarks
Chapter 9 Appendices
9.1 Controlled Approximation Problems
9.1.1 Setting of approximation problems
9.1.2 Solutions of approximation problems
9.1.3 Perturbation formulas
9.2 Weighted semigroups of composition operators
Bibliography
Subject Index
Author Index
Symbols
List of Figures