This chapter describes a technique for generating semi-abstract figurative imagery using variations on the Mandelbrot Set, evolved using a genetic algorithm. The Mandelbrot Set offers an infinite supply of complex fractal imagery, but its expressive ability is limited, as far as being a “material” for visual manipulation by artists. The technique described here achieves a unique and varied bran...

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...

In this paper we describe some features of the parameter planes for the families of rational maps given by Fλ(z) = z n + λ/zn where n ≥ 3, λ ∈ C. We assume n ≥ 3 since, in this case, there is a McMullen domain surrounding the origin in the λ-plane. This is a region where the corresponding Julia sets are Cantor sets of concentric simple closed curves. We prove here that the McMullen domain in th...

According to the method, suggested in our previous work (arxiv:nlin.CD/0509012) and based on the consideration of the specially coupled systems, the possibility of physical realization of the phenomena of complex analytic dynamics (such as Mandelbrot and Julia sets) is discussed. It is shown, that unlike the case of discrete maps or differential systems with periodic driving, investigated in me...

We give new proofs that the Mandelbrot set is locally connected at every Misiurewicz point and at every point on the boundary of a hyperbolic component. The idea is to show “shrinking of puzzle pieces” without using specific puzzles. Instead, we introduce fibers of the Mandelbrot set (see Definition 3.2) and show that fibers of certain points are “trivial”, i.e., they consist of single points. ...

Abstract. Due to its fractal nature, much about the area of the Mandelbrot set M remains to be understood. While a series formula has been derived by Ewing and Schober to calculate the area of M by considering its complement inside the Riemann sphere, to date the exact value of this area remains unknown. This paper presents new improved upper bounds for the area based on a parallel computing al...

We prove that given any θ1, . . . , θ2d−2 ∈ R\Z, the support of the bifurcation measure of the moduli space of degree d rational maps coincides with the closure of classes of maps having 2d − 2 neutral cycles of respective multipliers e1 , . . . , e2d−2 . To this end, we generalize a famous result of McMullen, proving that homeomorphic copies of (∂M) are dense in the support of the k-bifurcatio...