We give a new proof that all external rays of the Mandelbrot 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 different from the original one, given by Douady and Hubbard and refined by Lavaurs, in several ways: it replaces analytic arguments by combinatorial ones; it does not use complex analytic depen...

Using a family of higher degree polynomials as a bridge, together with complex surgery techniques, we construct a homeomorphism between any two limbs of the Mandelbrot set of common denominator. Induced by these homeomorphisms and complex conjugation we obtain an involution between each limb and itself, whose fixed points form a topological arc. All these maps have counterparts at the combinato...

Given a sequence of complex numbers ρn, we study the asymptotic distribution of the sets of parameters c ∈ C such that the quadratic maps z2+c has a cycle of period n and multiplier ρn. Assume 1 n log |ρn| → L. If L ≤ log 2, they equidistribute on the boundary of the Mandelbrot set. If L > log 2 they equidistribute on the equipotential of the Mandelbrot set of level 2L− 2 log 2. Introduction In...

We propose a numerical algorithm for the approximate computation of coefficients standardized Riemann map complement Mandelbrot set. Let $ f_c(z) = z^2+c be standard representation quadratic family and \mathcal{M} set \Psi : \overline{ \mathbb{C}}\setminus \mathbb{D}(0, 1)} \rightarrow \mathbb{C}} \setminus is normalized map. In introduced approach, we start with Fourier transform given formula...

The Julia set of the quadratic map fμ(z) = μz(1 − z) for μ not belonging to the Mandelbrot set is hyperbolic, thus varies continuously. It follows that a continuous curve in the exterior of the Mandelbrot set induces a continuous family of Julia sets. The focus of this article is to show that this family can be obtained explicitly by solving the initial value problem of a system of infinitely c...

Both the computer drawing of the complement of the Mandelbrot-like set of a one-parameter dependent complex exponential family of maps and the computer drawing of the Julia sets of the maps of this family, grow with the maximal number of iterations we choose. Some graphic examples of this growth, which evoke the image of a garden, are shown here.

The external arguments of the external rays theory of Douady and Hubbard is a valuable tool in order to analyze the Mandelbrot set, a typical case of discrete dynamical system used to study nonlinear phenomena. We suggest here a general method for the calculation of the external arguments of external rays landing at the hyperbolic components root points of the Mandelbrot set. Likewise, we prese...

Nonlocal neighborhood filters are modern and powerful techniques for image and signal denoising. In this paper, we give a probabilistic interpretation and analysis of the method viewed as a random walk on the patch space. We show that the method is intimately connected to the characteristics of diffusion processes, their escape times over potential barriers, and their spectral decomposition. In...