Multibrot Sets Research Articles

Singular Perturbations of Multibrot Set Polynomials

We will give a complete description of the dynamics of the rational map $N_{F_{M_c}}(z)=\frac{3z^4-2z^3+c}{4z^3-3z^2+c}$ where c is a complex parameter. These are rational maps $N_{F_{M_c}}$ arising from Newton's method. The polynomial of Newton iteration function is obtained from singularly perturbed of the Multibrot set polynomial.

Tricomplex Distance Estimation for Filled-In Julia Sets and Multibrot Sets

In this article, we present a distance estimation formula that can be used to ray trace 3D slices of the filled-in Julia sets and the Multibrot sets generated by the tricomplex polynomials of the form [Formula: see text] where [Formula: see text] is any integer greater than [Formula: see text].

Characterization of the Principal 3D Slices Related to the Multicomplex Mandelbrot Set

This article focuses on the dynamics of the different tridimensional principal slices of the multicomplex Multibrot sets. First, we define an equivalence relation between those slices. Then, we characterize them in order to establish similarities between their behaviors. Finally, we see that any multicomplex tridimensional principal slice is equivalent to a tricomplex slice up to an affine transformation. This implies that, in the context of tridimensional principal slices, Multibrot sets do not need to be generalized beyond the tricomplex space.

Taylor Coefficients of the Conformal Map for the Exterior of the Reciprocal of the Multibrot Set

Taylor Coefficients of the Conformal Map for the Exterior of the Reciprocal of the Multibrot Set

Epicycloids and Blaschke products

It is well known that the bounding curve of the central hyperbolic component of the Multibrot set in the parameter space of unicritical degree d polynomials is an epicycloid with cusps. The interior of the epicycloid gives the polynomials of the form which have an attracting fixed point. We prove an analogous result for unicritical Blaschke products: in the parameter space of degree d unicritical Blaschke products, the parabolic functions are parameterized by an epicycloid with cusps and inside this epicycloid are the parameters which give rise to maps having an attracting fixed point in the unit disk. We further study in more detail the case when in which every Blaschke product is unicritical in the unit disk.

TRICOMPLEX DYNAMICAL SYSTEMS GENERATED BY POLYNOMIALS OF ODD DEGREE

In this paper, we give the exact interval of the cross section of the Multibrot sets generated by the polynomial [Formula: see text] where [Formula: see text] and [Formula: see text] are complex numbers and [Formula: see text] is an odd integer. Furthermore, we show that the same Multibrots defined on the hyperbolic numbers are always squares. Moreover, we give a generalized 3D version of the hyperbolic Multibrot set and prove that our generalization is an octahedron for a specific 3D slice of the dynamical system generated by the tricomplex polynomial [Formula: see text] where [Formula: see text] is an odd integer.

Erratum to “The Decoration Theorem for Mandelbrot and Multibrot Sets”

Erratum to “The Decoration Theorem for Mandelbrot and Multibrot Sets”

The Decoration Theorem for Mandelbrot and Multibrot Sets

We prove the decoration theorem for the Mandelbrot (and Multibrot sets) which says that when a little Mandelbrot set is removed from the Mandelbrot set, then most of the resulting connected components have small diameters.

Cross-sections of multibrot sets

We identify the intersection of the multibrot set of \(z^d+c\) with the rays \({\mathbb R}^+\omega \), where \(\omega ^{d-1}=\pm 1\).

A remark on Zagier's observation of the Mandelbrot set

We investigate the arithmetic properties of the coefficients for the normalized conformal mapping of the exterior of the Multibrot set. In this paper, an estimate of the prime factor of the denominator of these coefficients is given. Bielefeld, Fisher and Haeseler [1] presented Zagier's observation for the growth of the denominator of the coefficients for the Mandelbrot set, and Yamashita [14] verified it. Our study takes into consideration both Zagier's observation and Yamashita's estimate.

A study of dynamics of the tricomplex polynomial $$\eta ^p+c$$ η p + c

In this article, we give the exact interval of the cross section of the so-called Mandelbric set generated by the polynomial \(z^3+c\) where \(z\) and \(c\) are complex numbers. Following that result, we show that the Mandelbric defined on the hyperbolic numbers \(\mathbb {D}\) is a square with its center at the origin. Moreover, we define the Multibrot sets generated by a polynomial of the form \(Q_{p,c}(\eta )=\eta ^p+c\) (\(p \in \mathbb {N}\) and \(p \ge 2\)) for tricomplex numbers. More precisely, we prove that the tricomplex Mandelbric has four principal slices instead of eight principal 3D slices that arise for the case of the tricomplex Mandelbrot set. Finally, we prove that one of these four slices is an octahedron.

Parapuzzle of the multibrot set and typical dynamics of unimodal maps

We study the parameter space of unicritical polynomials f_c : z ↦ z^d + c . For complex parameters, we prove that for Lebesgue almost every c , the map f_c is either hyperbolic or infinitely renormalizable. For real parameters, we prove that for Lebesgue almost every c , the map f_c is either hyperbolic, or Collet–Eckmann, or infinitely renormalizable. These results are based on controlling the spacing between consecutive elements in the “principal nest” of parapuzzle pieces.

Combinatorial rigidity for unicritical polynomials

We prove that any unicritical polynomial fc : z zd + c which is at most finitely renormalizable and has only repelling periodic points is combinatorially rigid. This implies that the connectedness locus (the Multibrot set) is locally connected at the corresponding parameter values and generalizes Yoccoz's Theorem for quadratics to the higher degree case.

A GENERALIZED VERSION OF THE MCMULLEN DOMAIN

We study the family of complex maps given by Fλ(z) = zn + λ/zn + c where n ≥ 3 is an integer, λ is an arbitrarily small complex parameter, and c is chosen to be the center of a hyperbolic component of the corresponding Multibrot set. We focus on the structure of the Julia set for a map of this form generalizing a result of McMullen. We prove that it consists of a countable collection of Cantor sets of closed curves and an uncountable number of point components.

A count of maximal small copies in Multibrot sets

We give a recursive formula to count maximal small copies of the Mandelbrot set and its higher degree analogues. This formula is used to compute the asymptotic growth of the number of maximal small copies of period n.