Complex Analysis and Dynamical Systems (eBook)

Preface 6
Contents 8
On Polynomially Integrable Domains in Euclidean Spaces 10
1 Introduction 10
2 Preliminaries 12
2.1 Support Functions 12
2.2 Radon Transform 13
3 There Are No Polynomially Integrable Domains in Even Dimensions 15
4 Polynomially Integrable Domains in R2m+1 Are Convex 17
5 Polynomially Integrable Domains in Rn, n=2m+1, of Degree ?n-1 Are Ellipsoids 20
6 Axially Symmetric Polynomially Integrable Domains in 3D Are Ellipsoids 24
7 Finite Stationary Phase Expansion Point of View 27
8 Open Questions 30
References 30
A Survey on the Maximal Number of Solutions of Equations Related to Gravitational Lensing 31
1 Introduction 31
2 Harmonic Polynomials, Rational Functions, and Lensing 33
2.1 Polynomials in z and barz 33
2.2 Harmonic Polynomials 34
2.3 Rational Functions and Gravitational Lensing 35
3 Gravitational Lensing: Continuous Mass Distributions 38
3.1 Ellipse with Uniform Mass Density 38
3.2 Investigating the Limaçon 39
3.2.1 The Lens Equation 39
3.2.2 The Schwarz Function 40
3.2.3 Solving the Lens Equation and Counting Solutions 41
3.3 Lens Equations Involving Transcendental Functions 44
References 45
Boundary Interpolation by Finite Blaschke Products 47
1 Introduction 47
2 The Modified Interpolation Problem 49
3 Parametrization of the Set of Low-Degree Solutions 51
4 Existence of Bn-2-Solutions 63
5 Examples 65
5.1 Three-Points Problem 65
5.2 Another Example 67
5.3 Boundary Fixed Points 68
6 Concluding Remarks and Open Questions 71
References 72
Support Points and the Bieberbach Conjecture in Higher Dimension 74
1 Introduction 74
2 The Class S0 76
3 Support Points and Extreme Points 78
4 Automorphisms of C2 and Support Points 79
5 Coefficient Bounds in B2 80
6 Operations in the Class M 82
6.1 Decoupling Harmonic Terms 82
6.2 Slice Reduction 82
7 Coefficient Bounds: q1m,0 and b1m,0 84
8 Coefficient Bounds: q10,m and b10,m 84
References 86
Some Unsolved Problems About Condenser Capacities on the Plane 87
1 Introduction 87
2 Generalized Condensers 88
3 Asymptotic Formulae 90
4 Extremal Decompositions 92
5 Set Capacities 93
6 A General Question 96
References 97
Filtration of Semi-complete Vector Fields Revisited 99
References 108
Polynomial Lemniscates and Their Fingerprints: From Geometry to Topology 109
1 Introduction 109
2 On the Geometry of Fingerprints 112
2.1 Real-Analytic Fingerprints 112
2.2 Shapes with Corners 113
2.3 Dynamics of Proper Lemniscates When They Are Approaching Critical Points 113
3 Nodes of Lemniscates and Inflection Points of Fingerprints 115
4 Polynomial Fireworks 120
4.1 Trees 121
4.2 Operad 122
4.3 Construction of Polynomial Fireworks 123
4.4 Operad Construction 126
References 134
Regularity of Mappings with Integrally Restricted Moduli 135
1 Introduction 135
2 Regularity Properties for General Homeomorphisms 136
3 Homeomorphisms with Integrally Restricted Moduli 137
4 Q-Homeomorphisms 138
5 Ring Q-Homeomorphisms 140
6 Lower Q-Homeomorphisms 144
References 145
Extremal Problems for Mappings with g-Parametric Representation on the Unit Polydisc in Cn 147
1 Introduction 148
2 Preliminaries 149
3 Extreme Points and Support Points for Mg(Un) and Sg*(Un) 154
4 Coefficient Bounds for the Family Mg(U2) 160
5 Bounded Support Points for the Family Sg0(U2) 166
6 Questions and Conjectures 170
References 172
Evolution of States of a Continuum Jump Model with Attraction 174
1 Introduction 174
1.1 Setup 174
1.2 Presenting the Result 175
2 Preliminaries and the Model 177
2.1 Configuration Spaces 177
2.2 Correlation Functions 179
2.3 The Model 180
3 The Result 181
3.1 The Operators 181
3.2 The Statement 184
3.3 Analyzing the Assumptions 185
3.4 Comments 186
4 The Proof 187
4.1 An Auxiliary Semigroup 187
4.2 Getting the Solutions 190
4.3 Proving the Uniqueness 193
References 194
Problems on Weighted and Unweighted Composition Operators 195
1 Introduction 195
2 Spectral Problems 198
2.1 Maps Fixing a Point in U 198
2.2 Maps Without Fixed Points in U 200
2.3 Complex Dynamics and Spectra 201
3 Numerical Ranges 202
4 Invariant Subspaces 204
4.1 The Invariant Subspace Problem via Composition Operators 204
4.2 Aleksandrov Operators 205
4.3 Description of the Invariant Subspace Lattice 207
4.4 Strongly Cyclic Composition Operators 208
5 Weighted Composition Operators 209
5.1 Boundedness and Compactness 209
5.2 Invertibility and Spectra 211
5.3 Normality Concepts 213
5.4 Hardy–Smirnov Spaces 213
5.5 Brennan's Conjecture and a Bergman Version of Hardy–Smirnov Spaces 215
6 Asymptotic Toeplitz Concepts 216
References 218
Harmonic Measures of Slit Sides, Conformal Welding and Extremum Problems 222
1 Introduction 222
2 Harmonic Measures of Slit Sides and Singular Solutions to the Loewner Equation 223
3 Conformal Mapping onto an Analytic Cusp 226
4 Asymptotic Conformal Welding via Loewner-Kufarev Evolution 228
5 The Bombieri Conjecture for Univalent Functions 230
References 232
Comparison Moduli Spaces of Riemann Surfaces 234
1 Introduction 234
2 Comparison Moduli Spaces 236
2.1 Moduli Space of Nested Surfaces 236
2.2 Moduli Spaces of Mappings 238
3 Some Examples in Geometric Function Theory 239
3.1 Nehari Monotonicity Theorems and Generalizations 239
3.2 A Few Remarks on the Extremal Metric Method 244
3.3 Schiffer Comparison Theory of Domains 245
3.4 Fredholm Determinant and Fredholm Eigenvalues 249
4 Teichmüller Space as a Comparison Moduli Space 250
4.1 Spaces of Non-overlapping Maps and the Rigged Moduli Space 250
4.2 A Brief Primer on Quasiconformal Teichmüller Theory 252
4.3 The Teichmüller/Rigged Moduli Space Correspondence 256
4.4 Some Applications of the Teichmüller Space/Rigged Moduli Space Correspondence 257
5 Weil-Petersson Class Teichmüller Space 258
5.1 The Weil-Petersson Class Teichmüller Space 258
5.2 Weil-Petersson Class Universal Teichmüller Space 260
5.3 Higher Genus Weil-Petersson Class Teichmüller Spaces 263
5.4 Kähler Potential of Weil-Petersson Metric 267
5.5 Applications of Weil-Petersson Class Teichmüller Theory 268
6 Conclusion 270
6.1 Concluding Remarks 270
References 270
Asymptotic Ratio of Harmonic Measures of Sides of a Boundary Slit 275
1 Main Results 276
2 Harmonic Measure and Convergence of Domains to the Kernel 279
3 Reflection Principle for Harmonic Measures 281
4 Radial Slit 282
5 Proof of Theorem 1 286
6 Proof of Theorem 2 289
7 Oscillating Slits 290
8 Further Questions 298
References 301
Coupling of Gaussian Free Field with General Slit SLE 302
1 Introduction 302
1.1 A Simple Example of Coupling 302
1.2 (?,?)-SLE Overview 305
1.3 Overview and Purpose of This Paper 307
2 Preliminaries 308
2.1 Vector Fields and Coordinate Transform 309
2.2 (?,?)-SLE Basics 310
2.3 Pre-pre-Schwarzian 316
2.4 Test Functions 318
2.5 Linear Functionals and Change of Coordinates 320
2.6 Fundamental Solution to the Laplace-Beltrami Equation 321
2.7 Gaussian Free Field 324
2.8 The Schwinger Functionals 327
3 Coupling Between SLE and GFF 329
4 Coupling of GFF with the Dirichlet Boundary Conditions 337
4.1 Chrodal SLE with Drift 341
4.2 Dipolar SLE with Drift 342
4.3 Radial SLE with Drift 344
4.4 General Remarks 347
4.5 Chordal SLE with Fixed Time Reparametrization 349
4.6 SLE with One Fixed Boundary Point 350
5 Coupling of GFF with Dirichlet-Neumann Boundary Conditions 352
6 Coupling of Twisted GFF 353
7 Perspectives 355
Appendix 1: Nature of Coupling 356
Appendix 2: Technical Remarks 359
Appendix 3: Some Formulas from Stochastic Calculus 364
References 366
A Marx-Strohhacker Type Result for Close-to-Convex Functions 368
1 Introduction and Preliminaries 368
2 Main Result 371
References 373