In this paper, we discuss several aspects of Julia sets, as well as those of the Mandelbrot set. We are interested in topological properties such as connectivity and local connectivity, geometric properties such as Hausdorff dimension and Lebesgue measure, and complex analytic properties such as holomorphic removability. As one can easily see from the pictures of numerical experiments, there is...

In this paper, we investigate the existence of extension operators fromW (Ω) toW (R) (1 < p < ∞) for a class of tree-shaped domains Ω with a self-similar fractal boundary previously studied by Mandelbrot and Frame. When the fractal boundary has no self-contact, the results of Jones imply that there exist such extension operators for all p ∈ [1,∞]. In the case when the fractal boundary self-inte...

The multiple-spiral medallions are beautiful decorations situated in the proximity of the small copies of the Mandelbrot set. They are composed by an infinity of babies Mandelbrot sets that have external arguments with known structure. In this paper we study the coupling patterns of the external arguments of the baby Mandelbrot sets in multiple-spiral medallions, that is, how these external arg...

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. Mathematics Subject Classification: 37F20

Abstract We generalize a combinatorial formula of Douady from the main cardioid to other hyperbolic components H Mandelbrot set, constructing an explicit piecewise linear map which sends set angles external rays landing on real axis.

An object is self-similar if it can be decomposed into smaller copies of itself (the structure of the whole is contained in its parts) (Mandelbrot 1982; Falconer 2003). Not all irregular shapes found in nature are necessarily fractals (Losa 2009). Monofractal objects are mainly characterized by four properties: a) irregularity of the shape; b) self-similarity of the structure; c) non-integer or...