| We give a new proof that all external rays of the Man-delbrot set at rational angles land, and of the relation between the external angle of such a ray and the dynamics at the landing point. Our proof is diierent from the original one, given by Douady and Hubbard and reened by Lavaurs, in several ways: it replaces analytic arguments by combinatorial ones; it does not use complex analytic depe...

A. – A decoration of the Mandelbrot set M is a part of M cut off by two external rays landing at some tip of a satellite copy of M attached to the main cardioid. In this paper we consider infinitely renormalizable quadratic polynomials satisfying the decoration condition, which means that the combinatorics of the renormalization operators involved is selected from a finite family of deco...

We continue the description of Mandelbrot and Multibrot sets and of Julia sets in terms of fibers which was begun in [S3] and [S4]. The question of local connectivity of these sets is discussed in terms of fibers and becomes the question of triviality of fibers. In this paper, the focus is on the behavior of fibers under renormalization and other surgery procedures. We show that triviality of f...

Fractals provide an innovative method for generating 3D images of real-world objects by using computational modelling algorithms based on the imperatives of selfsimilarity, scale invariance, and dimensionality. Of the many different types of fractals that have come into limelight since their origin, the family of Mandelbrot Set fractals has eluded both mathematicians and computer scientists ali...

We explore geometric properties of the Mandelbrot set $${{\mathcal {M}}}$$ , and corresponding Julia sets $${{\mathfrak {J}}}_c$$ near main cardioid. Namely, we establish that: (a) is locally connected at certain infinitely renormalizable parameters c bounded satellite type, providing first examples this kind; (b) The are also have positive area; (c) self-similar Siegel periodic type. approach ...

We investigate one-parameter analytic maps from the complex plane onto itself. We approximate the set of parameter values for the stable periodic orbits, which arise due to subsequent bifurcation from the period-1 orbit, with the aid of normal forms. This approximated set consists of a cactus of touching circles, whose sizes obey a very simple scaling law. From this scaling law the Hausdorff di...