Related papers: The escaping set of a quasiregular mapping

This article concerns the iteration of quasiregular mappings on $\mathbb{R}^d$ and entire functions on $\mathbb{C}$. It is shown that there are always points at which the iterates of a quasiregular map tend to infinity at a controlled rate.…

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…

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…

In this article, we investigate the boundary of the escaping set I(f) for quasiregular mappings on R^n, both in the uniformly quasiregular case and in the polynomial type case. The aim is to show that the boundary of I(f) is the Julia set…

We study, for the first time, the maximum modulus set of a quasiregular map. It is easy to see that these sets are necessarily closed, and contain at least one point of each modulus. Blumenthal showed that for entire maps these sets are…

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…

It is shown that for quasiregular maps of positive lower order the Julia set coincides with the boundary of the fast escaping set.

Let f be an entire function with a bounded set of singular values, and suppose furthermore that the postsingular set of f is bounded. We show that every component of the escaping set I(f) is unbounded. This provides a partial answer to a…

We investigate the rate of convergence of the iterates of an n-dimensional quasiregular mapping within the basin of attraction of a fixed point of high local index. A key tool is a refinement of a result that gives bounds on the distortion…

Let $f$ be a transcendental entire function. The escaping set $I(f)$ consists of those points that tend to infinity under iteration of $f$. We show that $I(f)$ is not $\sigma$-compact, resolving a question of Rippon from 2009.

Let $f$ be a transcendental entire function and let $I(f)$ denote the set of points that escape to infinity under iteration. We give conditions which ensure that, for certain functions, $I(f)$ is connected. In particular, we show that…

Let $f$ be a transcendental entire function. The fast escaping set $A(f)$, various regularity conditions on the growth of the maximum modulus of $f$, and also, more recently, the quite fast escaping set $Q(f)$ have all been used to make…

Let $f$ be a transcendental entire function and let $A(f)$ denote the set of points that escape to infinity `as fast as possible' under iteration. By writing $A(f)$ as a countable union of closed sets, called `levels' of $A(f)$, we obtain a…

Much recent work on the iterates of a transcendental entire function $f$ has been motivated by Eremenko's conjecture that all the components of the escaping set $I(f)$ are unbounded. Here we show that if $I(f)$ is disconnected, then the set…

Let $f$ be a transcendental entire function and $U$ be a Fatou component of $f$. We show that if $U$ is an escaping wandering domain of $f$, then most boundary points of $U$ (in the sense of harmonic measure) are also escaping. In the other…

We construct a quasiregular mapping in $\mathbb{R}^3$ that is the first to illustrate several important dynamical properties: the quasi-Fatou set contains wandering components; these quasi-Fatou components are bounded and hollow; and the…

Let $f$ and $g$ be two quasiregular maps in $\mathbb{R}^d$ that are of transcendental type and also satisfy $f\circ g =g \circ f$. We show that if the fast escaping sets of those functions are contained in their respective Julia sets then…

We give a new proof of the result that if f and g are transcendental entire functions, then the composite function f(g) has infinitely many fixed points. The method yields a number of generalization of this result. In particular, it extends…

We show that the points that converge to infinity under iteration of the exponential map form a connected subset of the complex plane.

The escaping set of an entire function consists of the points in the complex plane that tend to infinity under iteration. This set plays a central role in the dynamics of transcendental entire functions. The goal of this survey is to…