Related papers: The fast escaping set for quasiregular mappings

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

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…

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…

For many transcendental entire functions, the escaping set has the structure of a Cantor bouquet, consisting of uncountably many disjoint curves. Rippon and Stallard showed that there are many functions for which the escaping set has a new…

We introduce a new approximation technique into the context of complex dynamics that allows us to construct examples of transcendental entire functions with unbounded wandering domains. We provide examples of entire functions with an orbit…

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…

We give an example of a transcendental entire function with a simply connected fast escaping Fatou component, but with no multiply connected Fatou components. We also give a new criterion for points to be in the fast escaping set.

We construct a quasiregular analogue of the function $z\exp(z)$ in dimension 3, which gives the first explicit example of a quasiregular mapping of transcendental type that has exactly one zero. We then modify the construction to create a…

For a non-trivial transcendental semigroup, escaping set I(S) is in general S-forward invariant and it is S-completely invariant if semigroup S is abelian. In the contrary of this result, we investigate completely invariant escaping 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 show that, if the Julia set of a transcendental entire function is locally connected, then it takes the form of a spider's web in the sense defined by Rippon and Stallard. In the opposite direction, we prove that a spider's web Julia set…

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…

It is shown that the Hausdorff dimension of the fast escaping set of a quasiregular self-map of ${\mathbb R}^3$ can take any value in the interval $[1,3]$. The Hausdorff dimension of the Julia set of such a map is estimated under some…

We study a family of transcendental entire functions of genus zero, for which all of the zeros lie within a closed sector strictly smaller than a half-plane. In general these functions lie outside the Eremenko-Lyubich class. We show that…

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…

In this paper, we consider the Halpern iteration scheme for a finite family of quasinonexpansive mappings and then prove a strong convergence theorem to their common fixed point in a complete geodesic space with curvature bounded above by…

We mainly generalize the notion of abelian transcendental semigroup to nearly abelian transcendental semigroup. We prove that Fatou set, Julia set and escaping set of nearly abelian transcendental semigroup are completely invariant. We…

There are several classes of transcendental entire functions for which the Julia set consists of an uncountable union of disjoint curves each of which joins a finite endpoint to infinity. Many authors have studied the topological properties…

We wish to investigate some elementary problems concerning topological dynamics revolving around our proposed definition of escaping set. We also discuss the notion of escaping set in the induced dynamics of the hyperspace. Moreover, we…

Let $f$ and $g$ be transcendental entire functions, each with a bounded set of singular values, and suppose that $f$ and $g$ are affinely equivalent (that is, $g \circ \phi= \psi\circ f$, where $\phi,\psi:\C\to\C$ are affine). We show that…