In this paper, the dynamics of the Chebyshev-Halley family is studied on quadratic polynomials. A singular set, that we call cat set, appears in the parameter space associated to the family. This cat set has interesting similarities with the Mandelbrot set. The parameters space has allowed us to find different elements of the family such that can not converge to any root of the polynomial, sinc...

There is more and more empirical evidence that multifractality constitutes another and perhaps the most significant financial stylized fact. A realistic model of the financial dynamics should therefore incorporate this effect. The most promising in this respect is the Multifractal Model of Asset Returns (MMAR) introduced by Mandelbrot et al. [1] in which mul-tifractality is carried by time defo...

p1, p2, and p3 all being constants. The same inverse power-law statistical distributions were found in embarrassingly different situations (e.g., [6, 7]). In economics, it was discovered by Pareto [8] over 100 years ago and states that incomes of individuals or firms are inversely proportional to their rank. In less formal words [9], “most success seems to migrate to those people or companies w...

The multiple-spiral medallions are beautiful decorations of the Mandelbrot set. Computer graphics provide an invaluable tool to study the structure of these decorations with central symmetry, formed by an infinity of baby Mandelbrot sets that have high periods. Up to now, the external arguments of the external rays landing at the cusps of the cardioids of these baby Mandelbrot sets could not be...

By means of a graphical journey across the Mandelbrot set for the classic quadratic iterator f (z) : z 2 + q, we illustrate how connectivity breaks as the seed z 0 is no longer at the critical point of f (z). Finally we suggest an attack to the MLC conjecture.

We consider perturbations of the complex quadratic map $ z \to z^2 +c$ and corresponding changes in their quasi-Mandelbrot sets. Depending on particular perturbation, visual forms of quasi-Mandelbrot set changes either sharply (when the perturbation reaches some critical value) or continuously. In the latter case we have a smooth transition from the classical form of the set to some forms, cons...

In this note we inspect the kind of patterns produced by the quadratic iteration used to generate the Mandelbrot set. A very simple relation involving the Fibonacci numbers can be found between the relative positions of the disks which form the Mandelbrot set and the number of attractors of the iteration. Using this insight the concept of escape time can be generalized to points inside and not ...

The behavior at infinity of the Fourier transform of the random measures that appear in the theory of multiplicative chaos of Mandelbrot, Peyrière, and Kahane is an area quite unexplored. For context and further reference, we first present an overview of this theory and then the result, which is the main objective of this work, generalizing a result previously announced by Kahane. We establish ...