Agranovsky / Golberg / Jacobzon Complex Analysis and Dynamical Systems

New Trends and Open Problems
1. Auflage 2018
ISBN: 978-3-319-70154-7
Verlag: Springer International Publishing
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 373 Seiten

Reihe: Trends in Mathematics

This book focuses on developments in complex dynamical systems and geometric function theory over the past decade, showing strong links with other areas of mathematics and the natural sciences.Traditional methods and approaches surface in physics and in the life and engineering sciences with increasing frequency - the Schramm-Loewner evolution, Laplacian growth, and quadratic differentials are just a few typical examples. This book provides a representative overview of these processes and collects open problems in the various areas, while at the same time showing where and how each particular topic evolves. This volume is dedicated to the memory of Alexander Vasiliev.

Autoren/Hrsg.

Agranovsky, Mark

Golberg, Anatoly

Jacobzon, Fiana

Shoikhet, David

Zalcman, Lawrence

Inhaltsverzeichnis

1;Preface;6
2;Contents;8
3;On Polynomially Integrable Domains in Euclidean Spaces;10
3.1;1 Introduction;10
3.2;2 Preliminaries;12
3.2.1;2.1 Support Functions;12
3.2.2;2.2 Radon Transform;13
3.3;3 There Are No Polynomially Integrable Domains in Even Dimensions;15
3.4;4 Polynomially Integrable Domains in R2m+1 Are Convex;17
3.5;5 Polynomially Integrable Domains in Rn, n=2m+1, of Degree ?n-1 Are Ellipsoids;20
3.6;6 Axially Symmetric Polynomially Integrable Domains in 3D Are Ellipsoids;24
3.7;7 Finite Stationary Phase Expansion Point of View;27
3.8;8 Open Questions;30
3.9;References;30
4;A Survey on the Maximal Number of Solutions of Equations Related to Gravitational Lensing;31
4.1;1 Introduction;31
4.2;2 Harmonic Polynomials, Rational Functions, and Lensing;33
4.2.1;2.1 Polynomials in z and barz;33
4.2.2;2.2 Harmonic Polynomials;34
4.2.3;2.3 Rational Functions and Gravitational Lensing;35
4.3;3 Gravitational Lensing: Continuous Mass Distributions;38
4.3.1;3.1 Ellipse with Uniform Mass Density;38
4.3.2;3.2 Investigating the Limaçon;39
4.3.2.1;3.2.1 The Lens Equation;39
4.3.2.2;3.2.2 The Schwarz Function;40
4.3.2.3;3.2.3 Solving the Lens Equation and Counting Solutions;41
4.3.3;3.3 Lens Equations Involving Transcendental Functions;44
4.4;References;45
5;Boundary Interpolation by Finite Blaschke Products;47
5.1;1 Introduction;47
5.2;2 The Modified Interpolation Problem;49
5.3;3 Parametrization of the Set of Low-Degree Solutions;51
5.4;4 Existence of Bn-2-Solutions;63
5.5;5 Examples;65
5.5.1;5.1 Three-Points Problem;65
5.5.2;5.2 Another Example;67
5.5.3;5.3 Boundary Fixed Points;68
5.6;6 Concluding Remarks and Open Questions;71
5.7;References;72
6;Support Points and the Bieberbach Conjecture in Higher Dimension;74
6.1;1 Introduction;74
6.2;2 The Class S0;76
6.3;3 Support Points and Extreme Points;78
6.4;4 Automorphisms of C2 and Support Points;79
6.5;5 Coefficient Bounds in B2;80
6.6;6 Operations in the Class M;82
6.6.1;6.1 Decoupling Harmonic Terms;82
6.6.2;6.2 Slice Reduction;82
6.7;7 Coefficient Bounds: q1m,0 and b1m,0;84
6.8;8 Coefficient Bounds: q10,m and b10,m;84
6.9;References;86
7;Some Unsolved Problems About Condenser Capacities on the Plane;87
7.1;1 Introduction;87
7.2;2 Generalized Condensers;88
7.3;3 Asymptotic Formulae;90
7.4;4 Extremal Decompositions;92
7.5;5 Set Capacities;93
7.6;6 A General Question;96
7.7;References;97
8;Filtration of Semi-complete Vector Fields Revisited;99
8.1;References;108
9;Polynomial Lemniscates and Their Fingerprints: From Geometry to Topology;109
9.1;1 Introduction;109
9.2;2 On the Geometry of Fingerprints;112
9.2.1;2.1 Real-Analytic Fingerprints;112
9.2.2;2.2 Shapes with Corners;113
9.2.3;2.3 Dynamics of Proper Lemniscates When They Are Approaching Critical Points;113
9.3;3 Nodes of Lemniscates and Inflection Points of Fingerprints;115
9.4;4 Polynomial Fireworks;120
9.4.1;4.1 Trees;121
9.4.2;4.2 Operad;122
9.4.3;4.3 Construction of Polynomial Fireworks;123
9.4.4;4.4 Operad Construction;126
9.5;References;134
10;Regularity of Mappings with Integrally Restricted Moduli;135
10.1;1 Introduction;135
10.2;2 Regularity Properties for General Homeomorphisms;136
10.3;3 Homeomorphisms with Integrally Restricted Moduli;137
10.4;4 Q-Homeomorphisms;138
10.5;5 Ring Q-Homeomorphisms;140
10.6;6 Lower Q-Homeomorphisms;144
10.7;References;145
11;Extremal Problems for Mappings with g-Parametric Representation on the Unit Polydisc in Cn;147
11.1;1 Introduction;148
11.2;2 Preliminaries;149
11.3;3 Extreme Points and Support Points for Mg(Un) and Sg*(Un);154
11.4;4 Coefficient Bounds for the Family Mg(U2);160
11.5;5 Bounded Support Points for the Family Sg0(U2);166
11.6;6 Questions and Conjectures;170
11.7;References;172
12;Evolution of States of a Continuum Jump Model with Attraction;174
12.1;1 Introduction;174
12.1.1;1.1 Setup;174
12.1.2;1.2 Presenting the Result;175
12.2;2 Preliminaries and the Model;177
12.2.1;2.1 Configuration Spaces;177
12.2.2;2.2 Correlation Functions;179
12.2.3;2.3 The Model;180
12.3;3 The Result;181
12.3.1;3.1 The Operators;181
12.3.2;3.2 The Statement;184
12.3.3;3.3 Analyzing the Assumptions;185
12.3.4;3.4 Comments;186
12.4;4 The Proof;187
12.4.1;4.1 An Auxiliary Semigroup;187
12.4.2;4.2 Getting the Solutions;190
12.4.3;4.3 Proving the Uniqueness;193
12.5;References;194
13;Problems on Weighted and Unweighted Composition Operators;195
13.1;1 Introduction;195
13.2;2 Spectral Problems;198
13.2.1;2.1 Maps Fixing a Point in U;198
13.2.2;2.2 Maps Without Fixed Points in U;200
13.2.3;2.3 Complex Dynamics and Spectra;201
13.3;3 Numerical Ranges;202
13.4;4 Invariant Subspaces;204
13.4.1;4.1 The Invariant Subspace Problem via Composition Operators;204
13.4.2;4.2 Aleksandrov Operators;205
13.4.3;4.3 Description of the Invariant Subspace Lattice;207
13.4.4;4.4 Strongly Cyclic Composition Operators;208
13.5;5 Weighted Composition Operators;209
13.5.1;5.1 Boundedness and Compactness;209
13.5.2;5.2 Invertibility and Spectra;211
13.5.3;5.3 Normality Concepts;213
13.5.4;5.4 Hardy–Smirnov Spaces;213
13.5.5;5.5 Brennan's Conjecture and a Bergman Version of Hardy–Smirnov Spaces;215
13.6;6 Asymptotic Toeplitz Concepts;216
13.7;References;218
14;Harmonic Measures of Slit Sides, Conformal Welding and Extremum Problems;222
14.1;1 Introduction;222
14.2;2 Harmonic Measures of Slit Sides and Singular Solutions to the Loewner Equation;223
14.3;3 Conformal Mapping onto an Analytic Cusp;226
14.4;4 Asymptotic Conformal Welding via Loewner-Kufarev Evolution;228
14.5;5 The Bombieri Conjecture for Univalent Functions;230
14.6;References;232
15;Comparison Moduli Spaces of Riemann Surfaces;234
15.1;1 Introduction;234
15.2;2 Comparison Moduli Spaces;236
15.2.1;2.1 Moduli Space of Nested Surfaces;236
15.2.2;2.2 Moduli Spaces of Mappings;238
15.3;3 Some Examples in Geometric Function Theory;239
15.3.1;3.1 Nehari Monotonicity Theorems and Generalizations;239
15.3.2;3.2 A Few Remarks on the Extremal Metric Method;244
15.3.3;3.3 Schiffer Comparison Theory of Domains;245
15.3.4;3.4 Fredholm Determinant and Fredholm Eigenvalues;249
15.4;4 Teichmüller Space as a Comparison Moduli Space;250
15.4.1;4.1 Spaces of Non-overlapping Maps and the Rigged Moduli Space;250
15.4.2;4.2 A Brief Primer on Quasiconformal Teichmüller Theory;252
15.4.3;4.3 The Teichmüller/Rigged Moduli Space Correspondence;256
15.4.4;4.4 Some Applications of the Teichmüller Space/Rigged Moduli Space Correspondence;257
15.5;5 Weil-Petersson Class Teichmüller Space;258
15.5.1;5.1 The Weil-Petersson Class Teichmüller Space;258
15.5.2;5.2 Weil-Petersson Class Universal Teichmüller Space;260
15.5.3;5.3 Higher Genus Weil-Petersson Class Teichmüller Spaces;263
15.5.4;5.4 Kähler Potential of Weil-Petersson Metric;267
15.5.5;5.5 Applications of Weil-Petersson Class Teichmüller Theory;268
15.6;6 Conclusion;270
15.6.1;6.1 Concluding Remarks;270
15.7;References;270
16;Asymptotic Ratio of Harmonic Measures of Sides of a Boundary Slit;275
16.1;1 Main Results;276
16.2;2 Harmonic Measure and Convergence of Domains to the Kernel;279
16.3;3 Reflection Principle for Harmonic Measures;281
16.4;4 Radial Slit;282
16.5;5 Proof of Theorem 1;286
16.6;6 Proof of Theorem 2;289
16.7;7 Oscillating Slits;290
16.8;8 Further Questions;298
16.9;References;301
17;Coupling of Gaussian Free Field with General Slit SLE;302
17.1;1 Introduction;302
17.1.1;1.1 A Simple Example of Coupling;302
17.1.2;1.2 (?,?)-SLE Overview;305
17.1.3;1.3 Overview and Purpose of This Paper;307
17.2;2 Preliminaries;308
17.2.1;2.1 Vector Fields and Coordinate Transform;309
17.2.2;2.2 (?,?)-SLE Basics;310
17.2.3;2.3 Pre-pre-Schwarzian;316
17.2.4;2.4 Test Functions;318
17.2.5;2.5 Linear Functionals and Change of Coordinates;320
17.2.6;2.6 Fundamental Solution to the Laplace-Beltrami Equation;321
17.2.7;2.7 Gaussian Free Field;324
17.2.8;2.8 The Schwinger Functionals;327
17.3;3 Coupling Between SLE and GFF;329
17.4;4 Coupling of GFF with the Dirichlet Boundary Conditions;337
17.4.1;4.1 Chrodal SLE with Drift;341
17.4.2;4.2 Dipolar SLE with Drift;342
17.4.3;4.3 Radial SLE with Drift;344
17.4.4;4.4 General Remarks;347
17.4.5;4.5 Chordal SLE with Fixed Time Reparametrization;349
17.4.6;4.6 SLE with One Fixed Boundary Point;350
17.5;5 Coupling of GFF with Dirichlet-Neumann Boundary Conditions;352
17.6;6 Coupling of Twisted GFF;353
17.7;7 Perspectives;355
17.8;Appendix 1: Nature of Coupling;356
17.9;Appendix 2: Technical Remarks;359
17.10;Appendix 3: Some Formulas from Stochastic Calculus;364
17.11;References;366
18;A Marx-Strohhacker Type Result for Close-to-Convex Functions;368
18.1;1 Introduction and Preliminaries;368
18.2;2 Main Result;371
18.3;References;373