Related papers: The Julia sets of Chebyshev's method with small de…

We describe a rigorous computer algorithm for attempting to construct an explicit, discretized metric for which a complex polynomial map is expansive on a given neighborhood of its Julia set. We show construction of such a metric proves the…

We investigate the set of biaccessible points for connected polynomial Julia sets of arbitrary degrees $d\geq 2$. We prove that the Hausdorff dimension of the set of external angles corresponding to biaccessible points is less than 1,…

Iterative methods for the simultaneous determination of all roots of an equation are dis-cussed. The multiplicities of the roots are assumed to be known in advance. The methods are proved to have a cubical rate of convergence. Numerical…

We classify all solutions (p,q) to the equation p(u)q(u)=p(u+b)q(u+a) where p and q are complex polynomials in one indeterminate u, and a and b are fixed but arbitrary complex numbers. This equation is a special case of a system of…

We study polynomials with coefficients in a field L as dynamical systems where L is any algebraically closed and complete ultrametric field with dense valuation group and characteristic zero residual field. We give a complete description of…

Let $(f_\lambda)_{\lambda\in \Lambda}$ be a holomorphic family of polynomial automorphisms of $\mathbb{C}^2$. Following previous work of Dujardin and Lyubich, we say that such a family is weakly stable if saddle periodic orbits do not…

A generalization of the filled-in Julia set is presented using the multicomplex numbers and an algorithm is presented to visualize these sets in the tridimensional space. There are many ways to visualize these higher dimensional fractals…

Let $f$ be a rational map with degree at least two. We prove that $f$ has at least $2$ disjoint and infinite critical orbits in the Julia set if it has a Herman ring. This result is sharp in the following sense: there exists a cubic…

For a complex polynomial $P$ of degree $n$ and an $m$-tuple of distinct complex numbers $\Lambda=(\lambda_1,\ldots,\lambda_m)$, the dope matrix $D_P(\Lambda)$ is defined as the $m \times (n+1)$ matrix $(c)_{ij}$ with $c_{ij} =1$ if…

A matrix polynomial is a polynomial in a complex variable $\lambda$ with coefficients in $n \times n$ complex matrices. The spectral curve of a matrix polynomial $P(\lambda)$ is the curve $\{ (\lambda, \mu) \in \mathbb{C}^2 \mid…

For a class of polynomial maps of one variable with a parabolic fixed points and degrees bigger than $21$, the parabolic renormalization is introduced based on Fatou coordinates and horn maps, and a type of maps which are invariant under…

In this work, we study the non-autonomous dynamics generated by random iterations of the cubic family of the form $z^3 + cz$. The parameter sequence is chosen randomly from a bounded Borel subset of $\mathbb{C}$. We investigate topological…

We study the approximation of conformal mappings with the polynomials defined by Keldysh and Lavrentiev from an extremal problem considered by Julia. These polynomials converge uniformly on the closure of any Smirnov domain to the conformal…

We introduce a generalization of the McMullen family $f_{\lambda}(z)=z^n+\lambda/z^d$. In 1988, C. McMullen showed that the Julia set of $f_{\lambda}$ is a Cantor set of circles if and only if $1/n+1/d<1$ and the simple critical values of…

The theory of polynomial-like maps is of fundamental importance in holomorphic dynamics. We study dynamical properties of a larger class of maps. Our main result is that, under some natural conditions, a map of this class has a completely…

For a sequence of complex parameters $\{c_n\}$ we consider the compositions of functions $f_{c_n} (z) = z^2 + c_n$, which is the non-autonomous version of the classical quadratic dynamical system. The definitions of Julia and Fatou sets are…

We study complex one-dimensional parameter slices in a three-parameter family of rational maps with two free critical points, obtained by imposing the existence of periodic orbits with prescribed multipliers. Using explicit…

Let $p$ be a polynomial in the non-commuting variables $(a,x)=(a_1,...,a_{g_a},x_1,...,x_{g_x})$. If $p$ is convex in the variables $x$, then $p$ has degree two in $x$ and moreover, $p$ has the form $p = L + \Lambda ^T \Lambda,$ where $L$…

In this paper we introduce the notion of parabolic-like mapping, which is an object similar to a polynomial-like mapping, but with a parabolic external class, i.e. an external map with a parabolic fixed point. We prove a straightening…

We prove an extension results for the multiplier of an attracting periodic orbit of a quadratic map as a function of the parameter. This has applications to the problem of geometry of the Mandelbrot and Julia sets. In particular, we prove…