Showing papers on "External ray published in 1994"

Complex Dynamics and Renormalization

Abstract: Addressing researchers and graduate students in the active meeting ground of analysis, geometry, and dynamics, this book presents a study of renormalization of quadratic polynomials and a rapid introduction to techniques in complex dynamics. Its central concern is the structure of an infinitely renormalizable quadratic polynomial f(z) = z2 + c. As discovered by Feigenbaum, such a mapping exhibits a repetition of form at infinitely many scales. Drawing on universal estimates in hyperbolic geometry, this work gives an analysis of the limiting forms that can occur and develops a rigidity criterion for the polynomial f. This criterion supports general conjectures about the behavior of rational maps and the structure of the Mandelbrot set.The course of the main argument entails many facets of modern complex dynamics. Included are foundational results in geometric function theory, quasiconformal mappings, and hyperbolic geometry. Most of the tools are discussed in the setting of general polynomials and rational maps.

Internal addresses in the Mandelbrot set and irreducibility of polynomials

Abstract: We describe an interesting interplay between symbolic dynamics, the structure of the Mandelbrot set, permutations of periodic points achieved by analytic continuation, and Galois groups of certain polynomials. Internal addresses are a convenient and efficient way of describing the combinatorial structure of the Mandelbrot set, and of giving geometric meaning to the ubiquitous kneading sequences in human-readable form (Sections 3 and 4). A simple extension, \emph{angled internal addresses}, distinguishes combinatorial classes of the Mandelbrot set and in particular distinguishes hyperbolic components in a concise and dynamically meaningful way. This combinatorial description of the Mandelbrot set makes it possible to derive existence theorems for certain kneading sequences and internal addresses in the Mandelbrot set (Section~6) and to give an explicit description of the associated parameters. These in turn help to establish some algebraic results about permutations of periodic points and to determine Galois groups of certain polynomials (Section~7). Through internal addresses, various areas of mathematics are thus related in this manuscript, including symbolic dynamics and permutations, combinatorics of the Mandelbrot set, and Galois groups.

The ‘‘mystery’’ of the quadratic Mandelbrot set

Abstract: Numerical simulations suggest that the Mandelbrot set for binary numbers z=x+γy with x, y∈R and γ2=α+γβ (α,β∈R given) is a simple filled‐in square in the perplex case, i.e., α=1, β=0. The paper gives a proof of this experimental result.

Accessibility of typical points for invariant measures of positive Lyapunov exponents for iterations of holomorphic maps

Abstract: We prove that if A is the basin of immediate attraction to a periodic attracting or parabolic point for a rational map f on the Riemann sphere, ifA is completely invariant (i.e. f−1(A) = A), and if μ is an arbitrary f -invariant measure with positive Lyapunov exponents on ∂A, then μ-almost every point q ∈ ∂A is accessible along a curve from A. In fact, we prove the accessibility of every “good” q, i.e. one for which “small neighbourhoods arrive at large scale” under iteration of f . This generalizes the Douady–Eremenko–Levin–Petersen theorem on the accessibility of periodic sources. We prove a general “tree” version of this theorem. This allows us to deduce that on the limit set of a geometric coding tree (in particular, on the whole Julia set), if the diameters of the edges converge to 0 uniformly as the generation number tends to ∞, then every f -invariant probability ergodic measure with positive Lyapunov exponent is the image, via coding with the help of the tree, of an invariant measure on the full one-sided shift space. The assumption that f is holomorphic on A, or on the domain U of the tree, can be relaxed and one need not assume that f extends beyond A or U . Finally, we prove that if f is polynomial-like on a neighbourhood of C \A, then every “good” q ∈ ∂A is accessible along an external ray. Introduction. Let f : C→ C be a rational map of the Riemann sphere C. Let J(f) denote its Julia set. We say a periodic point p of period m is attracting (a sink) if |(fm)′(p)| 1 and parabolic if (fm)′(p) is a root of unity. We say that A = Ap is an immediate basin of attraction to a sink or a parabolic point p if A is a component of 1991 Mathematics Subject Classification: Primary 58F23. The author acknowledges the support by Polish KBN grants 210469101 “Iteracje i Fraktale” and 210909101 “Uklady Dynamiczne”. He would also like to thank the Institute of Mathematical Sciences of SUNY, Stony Brook, and the Institute of Mathematics of Yale University for their hospitality. The work on this paper was begun during his stays at these institutions in 1991/92.

On the complement of the Mandelbrot set

Abstract: We study the uniformization function of the Mandelbrot set via the behavior of multipliers of periodic orbits

The Julia Set of the Mapping z→zexp(z+μ)

Abstract: 1 Introduction and Main Results For an entire function f(z), we denote the n-th iteration of f by f~n. The Fatouset of f is defined by F(f)={z∈C|{f~n} is normal at z}, the complement J(f)=C\F(f)is called Julia set. A Julia set is a closed perfect set, and it is completelyinvariant under the mapping f.The iterated dynamical system of a complex analyticfunction was investigated by Fatou and Julia a long time ago. Recently, it hasbecome an active branch of the complex analysis.