Dynamics in one complex variable: introductory lectures

Abstract

These notes study the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. They are based on introductory lectures given at Stony Brook during the Fall Term of 1989-90. These lectures are intended to introduce the reader to some key ideas in the field, and to form a basis for further study. The reader is assumed to be familiar with the rudiments of complex variable theory and of two-dimensional differential geometry.

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• Thurston equivalence of topological polynomials
• The Julia sets and complex singularities in hierarchical Ising models
• Introduction to 1-summability and resurgence
• Infinitesimal Thurston Rigidity and the Fatou-Shishikura Inequality
• Rational parameter rays of the Mandelbrot set
• Rational rays and critical portraits of complex polynomials
• Rescaling limits of complex rational maps
• Computability of Brolin-Lyubich Measure
• Holomorphic Removability of Julia Sets
• On Fibers and Renormalization of Julia Sets and Multibrot Sets
• An answer to a question of Herman, Baker and Rippon concerning Siegel disks
• Characterization of the Principal 3D Slices Related to the Multicomplex Mandelbrot Set
• Expanding Thurston maps as quotients
• On Fibers and Local Connectivity of Compact Sets in C
• Holomorphic Motions and Normal Forms in Complex Analysis
• Modular forms, Schwarzian conditions, and symmetries of differential equations in physics
• Constructing an expanding metric for dynamical systems in one complex variable
• The Julia set of a post-critically finite endomorphism of PC^2
• Polynomial maps with a Julia set of positive measure
• The rigidity problem for analytic critical circle maps
• Regular Polynomial Endomorphisms of C^k
• Dynamics of the family lambda tan z
• Parameter scaling for the Fibonacci point
• Dynamical systems, simulation, abstract computation
• Filled Julia sets with empty interior are computable
• On the Size of Quadratic Siegel Disks: Part I
• Bernoulli measure of complex admissible kneading sequences
• Random Iteration of Rational Functions
• New Julia and Mandelbrot Sets for Jungck Ishikawa Iterates
• Rational Maps Whose Fatou Components Are Jordan Domains
• Dynamics of certain non-conformal degree two maps on the plane
• Toeplitz-composition C*-algebras for certain finite Blaschke products
• Surgery on postcritically finite rational maps by blowing up an arc
• Epsilon-neighborhoods of orbits and formal classification of parabolic diffeomorphisms
• Linearization in ultrametric dynamics in fields of characteristic zero - equal characteristic case
• Quotient algebras of Toeplitz-composition C*-algebras for finite Blaschke products
• Holomorphic Motions, Fatou Linearization, and Quasiconformal Rigidity for Parabolic Germs
• From an iteration formula to Poincaré's Isochronous Center Theorem for holomorphic vector fields
• Bifurcation currents in holomorphic dynamics on ${\mathbf{P}}^k$
• The numbers of periodic orbits of holomorphic mappings hidden at fixed points
• Constructing subdivision rules from rational maps
• Non-computable Julia sets
• Statistical properties of topological Collet-Eckmann maps
• Non-degenerate quadratic laminations
• Quasiperiodic orbits in Siegel disks/balls and the Babylonian problem
• Biaccessiblility in quadratic Julia sets II: The Siegel and Cremer cases
• Correspondence and translation principles for the Mandelbrot set
• On a solution to display non-filled-in quaternionic Julia sets
• Fixed point indices and periodic points of holomorphic mappings
• Wandering polygons and recurrent critical leaves
• Recurrence of entire transcendental functions with simple post singular sets
• Existence of the Ehresmann connection on a manifold foliated by the locally free action of a commutative Lie group
• Characterizing indecomposable plane continua from their complements
• Electrical networks on $n$-simplex fractals
• Regularization by free additive convolution, square and rectangular cases
• Perturbation of self-similar sets and some regular configurations and comparison of fractals
• Mating non-renormalizable quadratic polynomials
• Growth sequences for circle diffeomorphisms
• A connecting lemma for rational maps satisfying a no growth condition
• Computable Caratheodory Theory
• Non computable Mandelbrot-like set for a one-parameter complex family
• On hyperbolic attractors and repellers of endomorphisms
• Biaccessiblility in quadratic Julia sets I: The locally-connected case
• Rays to renormalizations
• New partially hyperbolic dynamical systems I
• Snowballs are Quasiballs
• Zeros of differential polynomials in real meromorphic functions
• Coding and tiling of Julia sets for subhyperbolic rational maps
• Return time statistics via inducing
• The Renormalization Method and Quadratic-Like Maps
• Local connectivity of the Mandelbrot set at certain infinitely renormalizable points
• Critical points on the boundaries of Siegel disks