Related papers: Julia sets for polynomial diffeomorphisms of ${\Bb…
A holomorphic endomorphism f of CP^2 admits a Julia set J_1, defined as usual to be the locus of non-normality of its iterates, and a (typically) smaller Julia set J_2, which is essentially the closure of the set of repelling periodic…
We show that if a meromorphic function has two completely invariant Fatou components and only finitely many critical and asymptotic values, then its Julia set is a Jordan curve. However, even if both domains are attracting basins, the Julia…
We prove that Julia components of polynomials are generally small in diameter. For polynomials without irrationally neutral cycles, Fatou components are also typically small, even when the Julia set is not locally connected.
In this article, we introduce the concept of normal families of bicomplex holomorphic functions to obtain a bicomplex Montel theorem. Moreover, we give a general definition of Fatou and Julia sets for bicomplex polynomials and we obtain a…
By means of theory group analysis, some algebraic and geometrical properties of quaternion analogs of \emph{Julia} sets are investigated. We argue that symmetries, intrinsic to quaternions, give rise to the class of identical \emph{Julia}…
The asymptotic behaviour of the solutions of Poincar\'e's functional equation $f(\lambda z)=p(f(z))$ ($\lambda>1$) for $p$ a real polynomial of degree $\geq2$ is studied in angular regions of the complex plain. The constancy of an occurring…
In this paper we settle most of the open questions on algorithmic computability of Julia sets. In particular, we present an algorithm for constructing quadratics whose Julia sets are uncomputable. We also show that a filled Julia set of a…
The goal of this paper is to study some basic properties of the Fatou and Julia sets for a family of holomorphic endomorphisms of $\mathbb{C}^k,\; k \ge 2$. We are particularly interested in studying these sets for semigroups generated by…
The long-standing problem of existence of nowhere dense rational Julia set with positive area has been solved by an example in quadratic polynomials by Buff and Ch\'eritat. Since then many efforts have been devoted to finding out new…
Let $f$ be a post-critically finite endomorphism (PCF map for short) on $\mathbb{P}^2$, let $J_1$ denote the Julia set and let $J_2$ denote the support of the measure of maximal entropy. In this paper we show that: 1. $J_1\setminus J_2$ is…
Holomorphic renormalization plays an important role in complex polynomial dynamics. We consider certain conditions guaranteeing that a polynomial which does not admit a polynomial-like connected Julia set still admits an invariant continuum…
The Hausdorff dimensions of the Julia sets for non-analytic maps: f(z) = z^2 + epsilon z^* and f(z) = {z^*}^2 + epsilon are calculated perturbatively for small epsilon. It is shown that Ruelle's formula for Hausdorff dimensions of analytic…
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)…
In this paper, we prove that escaping set of transcendental semigroup is S-forward invariant. We also prove that if holomorphic semigroup is abelian, then Fatou set, Julia set and escaping set are S-completely invariant. We see certain…
The no invariant line fields conjecture is one of the main outstanding problems in traditional complex dynamics. In this paper we consider non-autonomous iteration where one works with compositions of sequences of polynomials with suitable…
We give explicit examples of pairs of Julia sets of hyperbolic rational maps which are homeomorphic but not quasisymmetrically homeomorphic.
We study the dynamics of the family $f_c(x, y)= (xy+c, x)$ of endomorphisms of $\mathbb{R}^2$ and $\mathbb{C}^2$, where $c$ is a real or complex parameter. Such maps can be seen as perturbations of the map $f_0(x,y)=(xy,x)$, which is a…
Let $P$ be a polynomial of degree $d$ with a Cremer point $p$ and no repelling or parabolic periodic bi-accessible points. We show that there are two types of such Julia sets $J_P$. The \emph{red dwarf} $J_P$ are nowhere connected im…
In this work we study the backward filled Julia sets of a class of $p$-adic polynomial maps $f:\mathbb{Q}_p^2\longrightarrow \mathbb{Q}_p^2$ defined by $f(x,y)=(xy+c,x)$, where $c\in\mathbb{Q}_p$ is a $p$-adic number. In particular, if…
We investigate the dynamics of semigroups generated by a family of polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. Moreover, we investigate the associated random dynamics of polynomials.…