Related papers: Permutable Quasiregular Maps
We give three families of parabolic rational maps and show that every Cantor set of circles as the Julia set of a non-hyperbolic rational map must be quasisymmetrically equivalent to the Julia set of one map in these families for suitable…
In this paper we study, for the first time, Julia limiting directions of quasiregular mappings in $\mathbb{R}^n$ of transcendental-type. First, we give conditions under which every direction is a Julia limiting direction. Along the way, our…
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)…
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 show that for any quasimeromorphic mapping with an essential singularity at infinity, there exist points whose iterates tend to infinity arbitrarily slowly. This extends a result by Nicks for quasiregular mappings, and Rippon and…
We study the different rates of escape of points under iteration by holomorphic self-maps of $\mathbb C^*=\mathbb C\setminus\{ 0\}$ for which both 0 and $\infty$ are essential singularities. Using annular covering lemmas we construct…
Rational semigroups were introduced by Hinkkanen and Martin as a generalization of the iteration of a single rational map. There has subsequently been much interest in the study of rational semigroups. Quasiregular semigroups were…
In complex dynamics, the bungee set is defined as the set points whose orbit is neither bounded nor tends to infinity. In this paper we study, for the first time, the bungee set of a quasiregular map of transcendental type. We show that…
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…
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…
Uniformly quasiconformally homogeneous domains in $\mathbb{R}^n$ carry a transitive collection of $K$-quasiconformal maps for a fixed $K\geq 1.$ In this paper, we study two questions in this setting. The first is to show that…
The primary aim of this paper is to give topological obstructions to Cantor sets in $\mathbb{R}^3$ being Julia sets of uniformly quasiregular mappings. Our main tool is the genus of a Cantor set. We give a new construction of a genus $g$…
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…
Complex dynamics deals with the iteration of holomorphic functions. As is well- known, the first functions to be studied which gave non-trivial dynamics were quadratic polynomials, which produced beautiful computer generated pictures of…
C. Bishop has constructed an example of an entire function f in Eremenko-Lyubich class with at least two grand orbits of oscillating wandering domains. In this paper we show that his example has exactly two such orbits, that is, f has no…
This paper studies quasiconformal non-equivalence of Julia sets and limit sets. We proved that any Julia set is quasiconformally different from the Apollonian gasket. We also proved that any Julia set of a quadratic rational map is…
We show that wandering domains can exist in the Fatou set of a polynomial type quasiregular mapping of the plane. We also give an example of a quasiregular mapping of the plane, with an essential singularity at infinity, which has a…
Given commuting functions f,g, with at most a countable compact set of essential singularities, recent results for entire functions are extended to prove that Julia sets match, J(f)=J(g), in a particular case in the class K. With this…
We define commutator of a transcendental semigroup, and on the basis of this concept, we define conjugate semigroup. We prove that the conjugate semigroup is nearly abelian if and only if the given semigroup is nearly abelian. We also prove…
The object of the paper is to characterize gasket Julia sets of rational maps that can be uniformized by round gaskets. We restrict to rational maps without critical points on the Julia set. Under these conditions, we prove that a Julia set…