Julia Sets Of Rational Maps Research Articles

Existence and uniqueness of diffusions on the Julia sets of Misiurewicz-Sierpinski maps

We study the balanced resistance forms on the Julia sets of Misiurewicz-Sierpinski maps, which are self-similar resistance forms with equal weights. In particular, we use a theorem of Sabot to prove the existence and uniqueness of balanced forms on these Julia sets. We also provide an explorative study on the resistance forms on the Julia sets of rational maps with periodic critical points.

Quasisymmetric geometry of the Cantor circles as the Julia sets of rational maps

We give three families of parabolic rational maps and show that every Cantor set of circles as the Julia set of a non-hyperbolic rational map must be quasisymmetrically equivalent to the Julia set of one map in these families for suitable parameters. Combining a result obtained before, we give a complete classification of the Cantor circles Julia sets in the sense of quasisymmetric equivalence. Moreover, we study the regularity of the components of the Cantor circles Julia sets and establish a sufficient and necessary condition when a component of a Cantor circles Julia set is a quasicircle.

On the dimensions of Cantor Julia sets of rational maps

In this paper, we study the dimensions associated with the Cantor Julia set of a rational map whose Fatou set is an attracting domain. We prove that if the Julia set of such a map contains no persistently recurrent critical points, then the conformal dimension and the Hausdorff dimension of the Julia set are equal.

Wavelet Based Some Julia Sets of Rational Maps Having Zhukovskii Function

The dynamics of rational maps and their properties are interesting because of the presence of poles and zeros. In this paper we have computed Julia sets of rational maps having Zhukovskii Function for which the double of the first derivative has no Herman rings. The data points out of the Julia set in Matlab workspace were imported to Matlab Signal Processing Tool for their analysis. We have sampled the data points with the sampling frequency of 8192 Hz and obtained complex signals. We have then applied the band pass filter to these complex signals. The effect of the band pass filter has generated complex analogue modulated signals.

Extending external rays throughout the Julia sets of rational maps

For polynomial maps in the complex plane, the notion of external rays plays an important role in determining the structure of and the dynamics on the Julia set. In this paper we consider an extension of these rays in the case of rational maps of the form Fλ(z) = zn + λ/zn where n > 1. As in the case of polynomials, there is an immediate basin of ∞, so we have similar external rays. We show how to extend these rays throughout the Julia set in three specific examples. Our extended rays are simple closed curves in the Riemann sphere that meet the Julia set in a Cantor set of points and also pass through countably many Fatou components. Unlike the external rays, these extended rays cross infinitely many other extended rays in a manner that helps determine the topology of the Julia set.

Intertwined internal rays in Julia sets of rational maps

We show how the well-known concept of external rays in polynomial dynamics may be extended throughout the Julia set of certain rational maps. These new types of rays, which we call internal rays, meet the Julia set in a Cantor set of points, and each of t

Computability of Julia Sets

Among all computer-generated mathematical images, Julia sets of rational maps occupy one of the most prominent positions. Their beauty and complexity can be fascinating. They also hold a deep mathematical content. Computational hardness of Julia sets is the main subject of this book. By definition, a computable set in the plane can be visualized on a computer screen with an arbitrarily high magnification. There are countless programs to draw Julia sets. Yet, as the authors have discovered, it is possible to constructively produce examples of quadratic polynomials, whose Julia sets are not computable. This result is striking - it says that while a dynamical system can be described numerically with an arbitrary precision, the picture of the dynamics cannot be visualized. The book summarizes the present knowledge (most of it from the authors' own work) about the computational properties of Julia sets in a self-contained way. It is accessible to experts and students with interest in theoretical computer science or dynamical systems.

Dimensions of the Julia sets of rational maps with the backward contraction property

Consider a rational map f on the Riemann sphere of degree at least 2 which has no parabolic periodic points. Assuming that f has Rivera-Letelier's backward contraction property with an arbitrarily large constant, we show that the upper box dimension of the Julia set J(f) is equal to its hyperbolic dimension, by investigating the properties of conformal measures on the Julia set.

Disintegration of projective measures

In this paper, we study a class of quasi-invariant measures on paths generated by discrete dynamical systems. Our main result characterizes the subfamily of these measures which admit a certain disintegration. This is a disintegration with respect to a random walk Markov process which is indexed by the starting point of the paths. Our applications include wavelet constructions on Julia sets of rational maps on the Riemann sphere.

An explicit bound for uniform perfectness of the Julia sets of rational maps

A compact set C in the Riemann sphere is called uniformly perfect if there is a uniform upper bound on the moduli of annuli which separate C. Julia sets of rational maps of degree at least two are uniformly perfect. Proofs have been given independently by Ma ne and da Rocha and by Hinkkanen, but no explicit bounds are given. In this note, we shall provide such an explicit bound and, as a result, give another proof of uniform perfectness of Julia sets of rational maps of degree at least two. As an application, we provide a lower estimate of the Hausdorff dimension of the Julia sets. We also give a concrete bound for the family of quadratic polynomials $f_c(z)=z^2+c $ in terms of the parameter c.

Hausdorff dimension and conformal dynamics, III: Computation of dimension

This paper presents an eigenvalue algorithm for accurately computing the Hausdorff dimension of limit sets of Kleinian groups and Julia sets of rational maps. The algorithm is applied to Schottky groups, quadratic polynomials and Blaschke products, yielding both numerical and theoretical results. Dimension graphs are presented for (a) the family of Fuchsian groups generated by reflections in 3 symmetric geodesics; (b) the family of polynomials f c ( z ) = z 2 + c, c ∈ [-1, 1/2]; and (c) the family of rational maps f t ( z ) = z / t + 1/ z, t ∈ (0, 1]. We also calculate H.dim (Λ) ≈ 1.305688 for the Apollonian gasket, and H.dim ( J ( f )) ≈ 1.3934 for Douady's rabbit, where f ( z ) = z 2 + c satisfies f 3 (0) = 0.