Related papers: Julia sets of random exponential maps
We study topological properties of the escaping endpoints and fast escaping endpoints of the Julia set of complex exponential $\exp(z)+a$ when $a\in (-\infty,-1)$. We show neither space is homeomorphic to the whole set of endpoints. This…
We determine the exact Borel class of the points whose iterates under $\exp(z)+a$ tend to infinity. We also prove that the sets of non-escaping Julia points for many of these functions are topologically equivalent.
We prove that the set of all endpoints of the Julia set of $f(z)=\exp(z)-1$ which escape to infinity under iteration of $f$ is not homeomorphic to the rational Hilbert space $\mathfrak E$. As a corollary, we show that the set of all points…
Hereditarily non uniformly perfect (HNUP) sets were introduced by Stankewitz, Sugawa, and Sumi in \cite{SSS} who gave several examples of such sets based on Cantor set-like constructions using nested intervals. We exhibit a class of…
For maps of one complex variable, $f$, given as the sum of a degree $n$ power map and a degree $d$ polynomial, we provide necessary and sufficient conditions that the geometric limit as $n$ approaches infinity of the set of points that…
The characterization and properties of Julia sets of one parameter family of transcendental meromorphic functions $\zeta_\lambda(z)=\lambda \frac{z}{z+1} e^{-z}$, $\lambda >0$, $z\in \mathbb{C}$ is investigated in the present paper. It is…
There exist uniformly quasiregular maps $f:\mathbb{R}^3 \to \mathbb{R}^3$ whose Julia sets are wild Cantor sets.
The family of exponential maps $f_a(z)= e^z+a$ is of fundamental importance in the study of transcendental dynamics. Here we consider the topological structure of certain subsets of the Julia set $J(f_a)$. When $a\in (-\infty,-1)$, and more…
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}…
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…
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 investigate to what extent Fatou set, Julia set and escaping set of transcendental semigroup is respectively equal to the Fatou set, Julia set and escaping set of its subsemigroup. We define partial fundamental set and fundamental set of…
In this article, we study the dynamics of the following family of rational maps with one parameter: \begin{equation*} f_\lambda(z)= z^n+\frac{\lambda^2}{z^n-\lambda}, \end{equation*} where $n\geq 3$ and $\lambda\in\mathbb{C}^*$. This family…
Hereditarily non uniformly perfect (HNUP) sets were introduced by Stankewitz, Sugawa, and Sumi in \cite{SSS} who gave several examples of such sets based on Cantor set-like constructions using nested intervals. For non-autonomous iteration…
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…
We consider the family of transcendental entire maps given by $f_a(z)=a(z-(1-a))\exp(z+a)$ where $a$ is a complex parameter. Every map has a superattracting fixed point at $z=-a$ and an asymptotic value at $z=0$. For $a>1$ the Julia set of…
Conditions are given which imply that certain non-autonomous analytic iterated function systems (NIFS's) in the complex plane have uniformly perfect attractor sets, while other conditions imply the attractor is pointwise thin, and thus…
We show that for any transcendental meromorphic function $f$ there is a point $z$ in the Julia set of $f$ such that the iterates $f^n(z)$ escape, that is, tend to $\infty$, arbitrarily slowly. The proof uses new covering results for…
We introduce the concept of escaping set for semigroups of transcendental entire functions using Fatou-Julia theory. Several results of the escaping set associated with the iteration of one transcendental entire function have been extended…
Given a nondecreasing sequence $\Lambda=\{\lambda_n>0\}$ such that $\displaystyle\lim_{n\to\infty} \lambda_n=\infty,$ we consider the sequence $\mathcal N_\Lambda:=\left\{\lambda_ne^{i\theta_n},n\in\,\mathbb N\right\}$, where $\theta_n$ are…