Related papers: Slow escaping points of meromorphic functions
This paper continues our investigation of the dynamics of families of transcendental meromorphic functions with finitely many singular values all of which are finite. Here we look at a generalization of the family of polynomials…
Let $f$ and $g$ be permutable transcendental entire functions. We use a recent analysis of the dynamical behaviour in multiply connected wandering domains to make progress on the long standing conjecture that the Julia sets of $f$ and $g$…
We give a complete description of the possible Hausdorff dimensions of escaping sets for meromorphic functions with a finite number of singular values. More precisely, for any given $d\in [0,2]$ we show that there exists such a meromorphic…
We construct a transcendental entire $f:\mathbb{C}\rightarrow\mathbb{C}$ such that (1) $f$ has bounded singular set, (2) $f$ has a wandering domain, and (3) each singular value of $f$ escapes to infinity under iteration by $f$.
The fast escaping set of a transcendental entire function is the set of all points which tend to infinity under iteration as fast as compatible with the growth of the function. We study the analogous set for quasiregular mappings in higher…
Building on recent work by Rippon and Stallard, we explore the intricate structure of the spider's web fast escaping sets associated with certain transcendental entire functions. Our results are expressed in terms of the components of the…
We show that if the maximum modulus of a quasiregular mapping f grows sufficiently rapidly then there exists a non-empty escaping set I(f) consisting of points whose forward orbits under iteration tend to infinity. This set I(f) has an…
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…
This note initiates the study of the Fatou\,--\,Julia sets of a complex harmonic mapping. Along with some fundamental properties of the Fatou and the Julia sets, we observe some contrasting behaviour of these sets as those with in case of a…
We characterize the region of meromorphic continuation of an analytic function $f$ in terms of the geometric rate of convergence on a compact set of sequences of multi-point rational interpolants of $f$. The rational approximants have a…
A function which is transcendental and meromorphic in the plane has at least two singular values. On one hand, if a meromorphic function has exactly two singular values, it is known that the Hausdorff dimension of the escaping set can only…
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…
Let $f:\mathbb C\to \widehat{\mathbb C}=\mathbb C \cup\{\infty\}$ be a transcendental meromorphic function (possibly without any pole) with a single essential singularity, and that is chosen to be at $\infty$. The set of points…
Given two permutable entire functions $f$ and $g,$ we establish vital relationship between escaping sets of entire functions $f, g$ and their composition. We provide some families of transcendental entire functions for which Eremenko's…
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…
Let f be a real entire function whose set S(f) of singular values is real and bounded. We show that, if f satisfies a certain function-theoretic condition (the "sector condition"), then $f$ has no wandering domains. Our result includes all…
Let $f$ be a meromorphic function with bounded set of singular values and for which infinity is a logarithmic singularity. Then we show that $f$ has infinitely many repelling periodic points for any minimal period $n\geq1$, using a much…
The escaping set of an entire function is the set of points that tend to infinity under iteration. We consider subsets of the escaping set defined in terms of escape rates and obtain upper and lower bounds for the Hausdorff measure of these…
Structural stability of holomorphic functions has been the subject of much research in the last fifty years. Due to various technicalities, however, most of that work has focused on so-called finite-type functions (functions whose set of…
We investigate the existence of wandering Fatou components for polynomial skew-products in two complex variables. In 2004 the non-existence of wandering domains near a super-attracting invariant fiber was shown in [8]. In 2014 it was shown…