Related papers: Thick-Thin non-Autonomous Julia Sets
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…
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. The Julia set of such a semigroup may not be connected in general. We…
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…
For a sequence $(\lambda_n)$ of positive real numbers we consider the exponential functions $f_{\lambda_n} (z) = \lambda_n e^z$ and the compositions $F_n = f_{\lambda_n} \circ f_{\lambda_{n-1}} \circ ... \circ f_{\lambda_1}$. For such a…
We show that the Julia set of a non-elementary rational semigroup G is uniformly perfect when there is a uniform bound on the Lipschitz constants of the generators of G. This also proves that the limit set of a non-elementary Moebius group…
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 consider a sequence $(p_n)_{n=1}^\infty$ of polynomials with uniformly bounded zeros and $\deg p_1\geq 1$, $\deg p_n\geq 2$ for $n\geq 2$, satisfying certain asymptotic conditions. We prove that the function sequence $\left(\frac{1}{\deg…
We discuss the dynamic and structural properties of polynomial semigroups, a natural extension of iteration theory to random (walk) dynamics, where the semigroup $G$ of complex polynomials (under the operation of composition of functions)…
The goal of this article is to study a rigidity property of Julia sets of certain classes of automorphisms in $\mathbb{C}^k$, $k \ge 3.$ First, we study the relation between two polynomial shift-like maps in $\mathbb{C}^k$, $k \ge 3$, that…
For a hyperbolic polynomial automorphism of C^2 with a disconnected Julia set, and under a mild dissipativity condition, we give a topological description of the components of the Julia set. Namely, there are finitely many "quasi-solenoids"…
We investigate the dynamics of semigroups generated by 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.…
In this work we consider a class of endomorphisms of $\mathbb{R}^2$ defined by $f(x,y)=(xy+c,x)$, where $c\in\mathbb{R}$ is a real number and we prove that when $-1<c<0$, the forward filled Julia set of $f$ is the union of stable manifolds…
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.…
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…
It is well-known that the Julia set J(f) of a rational map is uniformly perfect; that is, every ring domain which separates J(f) has bounded modulus, with the bound depending only on f. In this article we prove that an analogous result is…
We study subgroups $H_U$ of the R. Thompson group $F$ which are stabilizers of finite sets $U$ of numbers in the interval $(0,1)$. We describe the algebraic structure of $H_U$ and prove that the stabilizer $H_U$ is finitely generated if and…
We extend a result regarding the Random Backward Iteration algorithm for drawing Julia sets (known to work for certain rational semigroups containing a non-M\"obius element) to a class of M\"obius semigroups which includes certain settings…
We study the possible Hausdorff limits of the Julia sets and filled Julia sets of subsequences of the sequence of dual Chebyshev polynomials of a non-polar compact set K in C and compare such limits to K. Moreover, we prove that the…
In this paper, we study rigidity of polynomials of arbitrary degree in the presence of neutral dynamics. Specifically, we focus on {non-renormalizable} (in the sense of Douady and Hubbard) complex polynomials of degree $d \geqslant 2$ that…
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…