Related papers: Iteration of meromorphic functions
This paper is devoted to establish sufficient conditions under which a transcendental meromorphic function has no unbounded Fatou components and to extend some results for entire functions to meromorphic functions. Actually, we shall mainly…
In this paper, we show that there exist transcendental meromorphic functions with a cycle of 2-periodic Fatou components, where one is simply connected while the other is doubly connected. In particular, the doubly connected Fatou component…
The Fatou-Julia theory for rational functions has been extended both to transcendental meromorphic functions and more recently to several different types of quasiregular mappings in higher dimensions. We extend the iterative theory to…
We completely characterise the bounded sets that arise as components of the Fatou and Julia sets of meromorphic functions. On the one hand, we prove that a bounded domain is a Fatou component of some meromorphic function if and only if it…
The dynamics of transcendental functions in the complex plane has received a significant amount of attention. In particular much is known about the description of Fatou components. Besides the types of periodic Fatou components that can…
We describe conditions under which a multiply connected wandering domain of a transcendental meromorphic function with a finite number of poles must be a Baker wandering domain, and we discuss the possible eventual connectivity of Fatou…
We study the iteration of transcendental self-maps of $\mathbb{C}^*:=\mathbb{C}\setminus \{0\}$, that is, holomorphic functions $f:\mathbb{C}^*\to\mathbb{C}^*$ for which both zero and infinity are essential singularities. We use…
This paper consists of tow parts. One is to study the existence of a point $a$ in the intersection of Julia set and escaping set such that $\arg z=\theta$ is a singular direction if $\theta$ is a limit point of $\{\arg f^n(a)\}$ under some…
We prove several results concerning the relative position of points in the postsingular set $P(f)$ of a meromorphic map $f$ and the boundary of a Baker domain or the successive iterates of a wandering component. For Baker domains we answer…
The Fatou-Julia iteration theory of rational functions has been extended to quasiregular mappings in higher dimension by various authors. The purpose of this paper is an analogous extension of the iteration theory of transcendental entire…
We first establish any continuum without interiors can be a limit set of iterations of an entire function on an oscillating wandering domain, and hence arise as a component of Julia sets. Recently, Luka Boc Thaler showed that every bounded…
We study the dynamics of an arbitrary semigroup of transcendental entire functions using Fatou-Julia theory. Several results of the dynamics associated with iteration of a transcendental entire function have been extended to transcendental…
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…
Let $M$ be the class of all transcendental meromorphic functions $f: \mathbb{C} \to \mathbb{C} \bigcup\{\infty\}$ with at least two poles or one pole that is not an omitted value, and $M_o =\{f \in M:f{has at least one omitted value}\}$.…
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 consider the dynamics associated with an arbitrary semigroup of transcendental entire functions. Fatou-Julia theory is used to investigate the dynamics of these semigroups. Several results of the dynamics associated with iteration of a…
In this paper, we prove that the ratio of the modulus of the iterates of two points in an escaping Fatou component may be bounded even if the orbit of the component contains an infinite modulus annulus sequence and this case cannot happen…
We show that if a meromorphic function has a direct singularity over infinity, then the escaping set has an unbounded component and the intersection of the escaping set with the Julia set contains continua. This intersection has an…
Let $f$ and $g$ be commuting meromorphic functions with finitely many poles. By studying the behaviour of Fatou components under this commuting relation, we prove that $f$ and $g$ have the same Julia set whenever $f$ and $g$ have no simply…
In this note it is shown that two key results on transcendental singularities for meromorphic functions of finite lower order have refinements which hold under the weaker hypothesis that the logarithmic derivative has finite lower order.