Iteration of rational functions : complex analytic dynamical systems / Alan F. Beardon

Includes bibliographical references (pages [273]-277) and indexes

Table Of Contents: Preface Prerequisites Examples Introduction Iteration of Mobius Transformations Iteration of z ↦ z2 Tchebychev Polynomials Iteration of z andmap z2-1 Iteration of z andmap z2 + c Iteration of z andmap z + 1/z Iteration of z andmap 2z - 1/z Newton's Approximation General Remarks Rational Maps The Extended Complex Plane Rational Maps The Lipschitz Condition Conjugacy Valency Fixed Points Critical Points A Topology on the Rational Functions The Fator and Julia Sets The Fatou and Julia Sets Completely Invariant Sets Normal Families and Equicontinuity The Hyperbolic Metric Properties of the Julia Set Exceptional Points Properties of the Julia Set Rational Maps with Empty Fatou Set Elliptic Functions The Structure of the Fatou Set The Toplogy of the Sphere Completely Invariant Components of the Fatou Set The Euler Characteristic The Riemann-Hurwitz Formula for Covering Maps Maps Between Components of the Fatou Set The Number of Components of the Fatou Set Components of the Julia Set Periodic Points The Classification of Periodic Points The Ecistence of Periodic Points (Super) Atracting Cycles Repelling Cycles Rationally Indifferent Cycles Irrationally Indifferent Cycles in F Irrationally Indifferent Cycles in J The Proof of the Existence of Periodic Points The Julia Set and Periodic Points Local Conjugacy Infinite Products The Universal Covering Surface Forward Invariant Components The Five Possibilities Limit Functions Parabolic Domains Siegel Discs and Herman Rings Connectivity of Invariant Components The No Wandering Domains Theorem The No Wandering Domains Theorem A Preliminary Result Conformal Structures Quasiconformal Conjugates of Rational Maps Boundary Values of Conjugate Maps The Proof of Theorem 8.1.2 Critical Points Introductory Remarks The Normality of Inverse Maps Critical Points and Periodic Domains Applications The Fatou Set of a Polynomial The Number of Non-Repelling Cycles Expanding Maps Julia Sets as Cantor Sets Julia Sets as Jordan Curves The Mandelbrot Set Hausdorff Dimension Hausdorff Dimension Computing Dimensions The Dimension of Julia Sets Examples Smooth Julia Sets Dendrites Components of F of Infinite Connectivity F with Infinitely Connected and Simply Connected Components J with Infinitely Many Non-Degenerate Components F of Infinite Connectivity with Critical Points in J A Finitely Connected Component of F J Is a Cantor Set of Circles The Function (z - 2)2/z2 References Index of Examples Index