Related papers: Chebyshev's method applied to polynomials with two…

We prove that a nonempty, proper subset $S$ of the complex plane can be approximated in a strong sense by polynomial filled Julia sets if and only if $S$ is bounded and $\hat{\mathbb{C}} \setminus \textrm{int}(S)$ is connected. The proof…

An inverse polynomial has a Chebyshev series expansion 1/\sum(j=0..k)b_j*T_j(x)=\sum'(n=0..oo) a_n*T_n(x) if the polynomial has no roots in [-1,1]. If the inverse polynomial is decomposed into partial fractions, the a_n are linear…

We prove local connectivity of the boundaries of invariant simply connected attracting basins for a class of transcendental meromorphic maps. The maps within this class need not be geometrically finite or in class $\mathcal B$, and the…

We develop a theory of quasisymmetries for finitely ramified fractals, with applications to finitely ramified Julia sets. We prove that certain finitely ramified fractals admit a naturally defined class of "undistorted metrics" that are all…

By using purely algebraic tools, we establish well-known properties of roots of Chebyshev polynomials. Especially, we show that these zeros are simple and lie in $(-1,1)$ and we prove in two ways that they are mostly irrational.

In this paper we generalize to bivariate polynomials of Fibonacci and Lucas, properties obtained for Chebyshev polynomials. We prove that the coordinates of the bivariate polynomials over appropriate basis are families of integers…

In a recent paper Lima, Panario and Wang have provided a new method to multiply polynomials in Chebyshev basis which aims at reducing the total number of multiplication when polynomials have small degree. Their idea is to use Karatsuba's…

A small perturbation of a quadratic polynomial with a non-repelling fixed point gives a polynomial with an attracting fixed point and a Jordan curve Julia set, on which the perturbed polynomial acts like angle doubling. However, there are…

In this paper we study a one parameter family of rational maps obtained by applying the Chebyshev-Halley root finding algorithms. We show that the dynamics near parameters where the family presents some degeneracy might be understood from…

Recursive algebraic construction of two infinite families of polynomials in $n$ variables is proposed as a uniform method applicable to every semisimple Lie group of rank $n$. Its result recognizes Chebyshev polynomials of the first and…

Any finite union of disjoint, mutually exterior Jordan curves in the complex plane can be approximated arbitrarily well in the Hausdorff topology by polynomial Julia sets. Furthermore, the proof is constructive.

By a symmetry of the Julia set of a polynomial, also referred as polynomial Julia set, we mean an Euclidean isometry preserving the Julia set. Each such symmetry is in fact a rotation about the centroid of the polynomial. In this article, a…

This paper is a continuation of authors work: Fatou and Julia like sets,Ukranian J. Math., to appear/arXiv:2006.08308[math.CV](see [4]). Here, we introduce escaping like set and generalized escaping like set for a family of holomorphic…

We prove the existence of quadratic polynomials having a Julia set with positive Lebesgue measure in three cases: the presence of a Cremer fixed point, the presence of a Siegel disk, the presence of infinitely many (satellite)…

We study the dynamics of polynomial maps on the boundary of the central hyperbolic component $\mathcal H_d$. We prove the local connectivity of Julia sets and a rigidity theorem for maps on the regular part of $\partial\mathcal H_d$. Our…

We give a direct and intuitive proof (via sliding some columns up and down) of the following interesting fact: if we write out the Chebyshev polynomials in a chart and take the sums of coefficients along certain diagonals, we obtain the…

The paper deals with a special filtered approximation method, which originates interpolation polynomials at Chebyshev zeros by using de la Vall\'ee Poussin filters. These polynomials can be an useful device for many theoretical and…

This article gives a precise description of the Fatou sets and Julia sets of matrix-valued polynomials in $\mathcal{M}(2,\mathbb{C})$ in terms of the corresponding polynomials in $\mathbb{C}$. Further, we construct Green functions and…

We derive two formulas for the weighted sums of rooted spanning forests of particular sequence of graphs by using the matrix tree theorem. We consider cycle graphs with edges so called the pendant edges. One of our formula can be described…

We propose a new method of calculation of generating functions of Chebyshev polynomials in several variables associated with root systems of simple Lie algebras. We obtain the generating functions of the polynomials in two variables…