Julia set

Abstract: Part One: One-Dimensional Dynamics Examples of Dynamical Systems Preliminaries from Calculus Elementary Definitions Hyperbolicity An example: the quadratic family An Example: the Quadratic Family Symbolic Dynamics Topological Conjugacy Chaos Structural Stability Sarlovskiis Theorem The Schwarzian Derivative Bifurcation Theory Another View of Period Three Maps of the Circle Morse-Smale Diffeomorphisms Homoclinic Points and Bifurcations The Period-Doubling Route to Chaos The Kneeding Theory Geneaology of Periodic Units Part Two: Higher Dimensional Dynamics Preliminaries from Linear Algebra and Advanced Calculus The Dynamics of Linear Maps: Two and Three Dimensions The Horseshoe Map Hyperbolic Toral Automorphisms Hyperbolicm Toral Automorphisms Attractors The Stable and Unstable Manifold Theorem Global Results and Hyperbolic Sets The Hopf Bifurcation The Hnon Map Part Three: Complex Analytic Dynamics Preliminaries from Complex Analysis Quadratic Maps Revisited Normal Families and Exceptional Points Periodic Points The Julia Set The Geometry of Julia Sets Neutral Periodic Points The Mandelbrot Set An Example: the Exponential Function

Abstract: Iterated function systems (i. f. ss) are introduced as a unified way of generating a broad class of fractals. These fractals are often attractors for i. f. ss and occur as the supports of probability measures associated with functional equations. The existence of certain ‘ p -balanced’ measures for i. f. ss is established, and these measures are uniquely characterized for hyperbolic i. f. ss. The Hausdorff—Besicovitch dimension for some attrac­tors of hyperbolic i. f. ss is estimated with the aid of p -balanced measures. What appears to be the broadest framework for the exactly computable moment theory of p -balanced measures — that of linear i. f. ss and of probabilistic mixtures of iterated Riemann surfaces — is presented. This extensively generalizes earlier work on orthogonal polynomials on Julia sets. An example is given of fractal reconstruction with the use of linear i. f. ss and moment theory.

Abstract: This paper attempts to describe some of the results obtained in the iteration theory of transcendental meromorphic functions, not excluding the case of entire functions. The reader is not expected to be familiar with the iteration theory of rational functions. On the other hand, some aspects where the transcendental case is analogous to the rational case are treated rather briefly here. For example, we introduce the different types of components of the Fatou set that occur in the iteration of rational functions but omit a detailed description of these types. Instead, we concentrate on the types of components that are special to transcendental functions (Baker domains and wandering domains).