A count of maximal small copies in Multibrot sets

Journal of Difference Equations and Applications, 2016

We consider the family of rational maps given by F λ (z) = z n +λ/z d 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 Sierpiński holes. Sierpiński holes are open simply connected subsets of the parameter space for which the Julia sets of F λ are Sierpiński curves. In this paper we generalize the symmetry properties of F λ 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.

We use a commutative generalization of complex numbers called bicomplex numbers to introduce the bicomplex dynamics of polynomials of Rochon [9] proved that the Mandelbrot set of quadratic polynomials in bicomplex numbers of the form w 2 + c is connected. We prove that our generalized Mandelbrot set of polynomials of type E d , f c (w) = w(w + c) d , is connected.

The present paper is motivated from the paper of John R. Tippetts (Tippetts 1992) in which he gave an algorithm to generate an interesting Mandelbrot set. We not only generate Julia sets using Tippetts algorithm (Tippetts 1992), but also generate some new Julia and Mandelbrot sets by slightly modifying the Tippetts algorithm. This approach yields a new class of algorithms to produce new and alluring fractals with virtually infinite complexity.

Nonlinearity, 2008

We consider self-similar sets in the plane for which a cyclic group acts transitively on the pieces. Examples like n-gon Sierpiński gaskets, Gosper snowflake and terdragon are well-known, but we study the whole family. For each n our family is parametrized by the points in the unit disk. Due to a connectedness criterion, there are corresponding Mandelbrot sets which are used to find various new examples with interesting properties. The Mandelbrot sets for n > 2 are regular-closed, and the open set condition holds for all parameters on their boundary, which is not known for the case n = 2.

Chaos, Solitons & Fractals: X, 2019

We provide an analytic estimate for the size of the bulbs adjoining the main cardioid of the Mandelbrot set. The bulbs are approximate circles, and are associated with the stability regions in the complex parameter μ-space of period-q orbits of the underlying map z → z 2 − μ. For the (p , q) orbit with winding number p / q , the associated stability bulb is an approximate circle with radius 1 q 2 sin π p q .

Chaos: An Interdisciplinary Journal of Nonlinear Science, 1998

We graphically study the Mandelbrot-like set of the complex exponential family of maps

The fractals generated from the self-squared function, 2 z z c  where z and c are complex quantities have been studied extensively in the literature. This paper studies the transformation of the function ,2 n z z c n    and analyzed the z plane and c plane fractal images generated from the iteration of these functions using Ishikawa iteration for integer and non-integer values.

Chaos, Solitons & Fractals, 2007

We show here in a graphic and simple way the relation between the periodic and chaotic regions in the Mandelbrot set. Since the relation between the periodic and chaotic regions in a onedimensional (1D) quadratic set is already well known, we shall base on it to extend the results to the Mandelbrot set. We shall see that in the same way as the hyperbolic components of the perioddoubling cascade determines the chaotic bands structure in 1D quadratic sets, the periodic region determines the chaotic region in the Mandelbrot set.

Probability and Mathematical Statistics

We develop, in the context of the boundary of a supercritical Galton–Watson tree, a uniform version of the argument used by Kahane 1987 on homogeneous trees to estimate almost surely and simultaneously the Hausdorff and packing dimensions of the Mandelbrot measure over a suitable set J . As an application, we compute, almost surely and simultaneously, the Hausdorff and packing dimensions of the level sets Eα of infinite branches of the boundary of the tree along which the averages of the branching random walk have a given limit point.

Journal of Mathematical Analysis and Applications, 1992

A useful formula is given for the coefficients of the conformal mapping from the unit disk onto the complement of the reciprocal of the Mandelbrot set. As a consequence, a number of coefficients are determined. Finally, a conjecture is made that would imply the local connectedness of the boundary of the Mandelbrot set.