Showing papers on "External ray published in 2017"

A landing theorem for entire functions with bounded post-singular sets.

Abstract: The Douady-Hubbard landing theorem for periodic external rays is one of the cornerstones of the study of polynomial dynamics. It states that, for a complex polynomial with bounded postcritical set, every periodic external ray lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic point is the landing point of at least one periodic external ray. We prove an analogue of this theorem for an entire function with bounded postsingular set. If the function has finite order of growth, then it is known that the escaping set contains certain curves called "periodic hairs"; we show that every periodic hair lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic hair. For a postsingularly bounded entire function of infinite order, such hairs may not exist. Therefore we introduce certain dynamically natural connected sets, called "dreadlocks". We show that every periodic dreadlock lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic dreadlock. More generally, we prove that every point of a hyperbolic set is the landing point of a dreadlock.

Generalized baby Mandelbrot sets adorned with halos in families of rational maps

Abstract: We consider the family of rational maps given by Fλ(z)=zn+λ/zd where n,d∈N with 1/n+1/d<1, the variable z∈C^ and the parameter λ∈C. It is known [1] that when n=d≥3 there are n-1 small copies of the Mandelbrot set symmetrically located around the origin in the parameter λ-plane. These baby Mandelbrot sets have ‘antennas’ attached to the boundaries of Sierpinski holes. Sierpinski holes are open simply connected subsets of the parameter space for which the Julia sets of Fλ are Sierpinski curves. In this paper we generalize the symmetry properties of Fλ and the existence of the n-1 baby Mandelbrot sets to the case when 1/n+1/d<1 where n is not necessarily equal to d.

Models for spaces of dendritic polynomials

Abstract: Complex 1-variable polynomials with connected Julia sets and only repelling periodic points are called \emph{dendritic}. By results of Kiwi, any dendritic polynomial is semi-conjugate to a topological polynomial whose topological Julia set is a dendrite. We construct a continuous map of the space of all cubic dendritic polynomials onto a laminational model that is a quotient space of a subset of the closed bidisk. This construction generalizes the "pinched disk" model of the Mandelbrot set due to Douady and Thurston. It can be viewed as a step towards constructing a model of the cubic connectedness locus.

An effective algorithm to compute Mandelbrot sets in parameter planes

Abstract: In McMullen (2000) it was proven that copies of generalized Mandelbrot set are dense in the bifurcation locus for generic families of rational maps. We develop an algorithm to an effective computation of the location and size of these generalized Mandelbrot sets in parameter space. We illustrate the effectiveness of the algorithm by applying it to concrete families of rational and entire maps.