Meromorphic Dynamics: Volume 2 : Elliptic Functions with an Introduction to the Dynamics of Meromorphic Functions

Cover
Half-title
Series information
Title page
Copyright information
Dedication
Contents
Preface
Acknowledgments
Introduction
Part III Topological Dynamics of Meromorphic Functions
13 Fundamental Properties of Meromorphic Dynamical Systems
13.1 Basic Iteration of Meromorphic Functions
13.2 Classification of Periodic Fatou Components
13.3 The Singular Sets Sing(f[sup(-n)]), Asymptotic Values, and Analytic Inverse Branches
14 Finer Properties of Fatou Components
14.1 Properties of Periodic Fatou Components
14.2 Simple Connectedness of Fatou Components
14.3 Baker Domains
14.4 Fatou Components of Class [mathcal(B)] and [mathcal(S)] of Meromorphic Functions
15 Rationally Indifferent Periodic Points
15.1 Local and Asymptotic Behavior of Analytic Functions Locally Defined Around Rationally Indifferent Fixed Points
15.2 Leau–Fatou Flower Petals
15.3 Fatou Flower Theorem and Fundamental Domains Around Rationally Indifferent Periodic Points
15.4 Quantitative Behavior of Analytic Functions Locally Defined Around Rationally Indifferent Periodic Points: Conformal Measures Outlook
Part IV Elliptic Functions: Classics, Geometry, and Dynamics
16 Classics of Elliptic Functions: Selected Properties
16.1 Periods, Lattices, and Fundamental Regions
16.2 General Properties of Elliptic Functions
16.3 Weierstrass ℘-Functions I
16.4 The Field of Elliptic Functions
16.5 The Discriminant of a Cubic Polynomial
16.6 Weierstrass ℘-Functions II
17 Geometry and Dynamics of (All) Elliptic Functions
17.1 Forward and Inverse Images of Open Sets and Fatou Components
17.2 Fundamental Structure Results
17.3 Hausdorff Dimension of Julia Sets of (General) Elliptic Functions
17.4 Elliptic Function as a Member of [mathcal(A)](X) for Forward Invariant Compact Sets X ⊆ [mathbb(C)]
17.5 Radial Subsets of J(f) and Various Dynamical Dimensions for Elliptic Functions f : [mathbb(C)] → [widehat(mathbb(C))]
17.6 Sullivan Conformal Measures for Elliptic Functions
17.7 Hausdorff Dimension of Escaping Sets of Elliptic Functions
17.8 Conformal Measures of Escaping Sets of Elliptic Functions
Part V Compactly Nonrecurrent Elliptic Functions: First Outlook
18 Dynamics of Compactly Nonrecurrent Elliptic Functions
18.1 Fundamental Properties of Nonrecurrent Elliptic Functions: Mañé's Theorem
18.2 Compactly Nonrecurrent Elliptic Functions: Definition, Partial Order in Crit[sub(c)](J(f)), and Stratification of Closed Forward-Invariant Subsets of J(f)
18.3 Holomorphic Inverse Branches
18.4 Dynamically Distinguished Classes of Elliptic Functions
19 Various Examples of Compactly Nonrecurrent Elliptic Functions
19.1 The Dynamics of Weierstrass Elliptic Functions: Some Selected General Facts
19.2 The Dynamics of Square Weierstrass Elliptic Functions: Some Selected Facts
19.3 The Dynamics of Triangular Weierstrass Elliptic Functions: Some Selected Facts
19.4 Simple Examples of Dynamically Different Elliptic Functions
19.5 Expanding (Thus Compactly Nonrecurrent) Triangular Weierstrass Elliptic Functions with Nowhere Dense Connected Julia Sets
19.6 Triangular Weierstrass Elliptic Functions Whose Critical Values Are Preperiodic, Thus Being Subexpanding
19.7 Weierstrass Elliptic Functions Whose Critical Values Are Poles or Prepoles, Thus Being Subexpanding, Thus Compactly Nonrecurrent
19.8 Compactly Nonrecurrent Elliptic Functions with Critical Orbits Clustering at Infinity
19.9 Further Examples of Compactly Nonrecurrent Elliptic Functions
Part VI Compactly Nonrecurrent Elliptic Functions: Fractal Geometry, Stochastic Properties, and Rigidity
20 Sullivan h-Conformal Measures for Compactly Nonrecurrent Elliptic Functions
20.1 Existence of Conformal Measures for Compactly Nonrecurrent Elliptic Functions
20.2 Conformal Measures for Compactly Nonrecurrent Elliptic Functions and Holomorphic Inverse Branches
20.3 Conformal Measures for Compactly Nonrecurrent Regular Elliptic Functions: Atomlessness, Uniqueness, Ergodicity, and Conservativity
21 Hausdorff and Packing Measures of Compactly Nonrecurrent Regular Elliptic Functions
21.1 Hausdorff Measures
21.2 Packing Measure I
21.3 Packing Measure II
22 Conformal Invariant Measures for Compactly Nonrecurrent Regular Elliptic Functions
22.1 Conformal Invariant Measures for Compactly Nonrecurrent Regular Elliptic Functions: The Existence, Uniqueness, Ergodicity/Conservativity, and Points of Finite Condensation
22.2 Real Analyticity of the Radon–Nikodym Derivative [frac(dμ[sub(h))(dm[sub(h))]
22.3 Finite and Infinite Condensation of Parabolic Periodic Points with Respect to the Invariant Conformal Measure μ[sub(h)]
22.4 Closed Invariant Subsets, K(V) Sets, and Summability Properties
22.5 Normal Subexpanding Elliptic Functions of Finite Character: Stochastic Properties and Metric Entropy, Young Towers, and Nice Sets Techniques
22.6 Parabolic Elliptic Maps: Nice Sets, Graph Directed Markov Systems, Conformal and Invariant Measures, Metric Entropy
22.7 Parabolic Elliptic Maps with Finite Invariant Conformal Measures: Statistical Laws, Young Towers, and Nice Sets Techniques
22.8 Infinite Conformal Invariant Measures: Darling–Kac Theorem for Parabolic Elliptic Functions
23 Dynamical Rigidity of Compactly Nonrecurrent Regular Elliptic Functions
23.1 No Compactly Nonrecurrent Regular Function is Esentially Linear
23.2 Proof of the Rigidity Theorem
Appendix A A Quick Review of Some Selected Facts from Complex Analysis of a One-Complex Variable
Appendix B Proof of the Sullivan Nonwandering Theorem for Speiser Class [mathcal(S)]
References
Index of Symbols
Subject Index