Related papers: Annular itineraries for entire functions

Let $f$ be a transcendental entire function. It was shown in a previous paper that the holomorphic flow $\dot z = f(z)$ always has infinitely many trajectories tending to infinity in finite time. It will be proved here that such…

A transcendental entire function is called criniferous if every point in its escaping set can eventually be connected to infinity by a curve of escaping points. Many transcendental entire functions with bounded singular set have this…

In 1902, P. St\"{a}ckel proved the existence of a transcendental function $f(z)$, analytic in a neighbourhood of the origin, and with the property that both $f(z)$ and its inverse function assume, in this neighbourhood, algebraic values at…

In 1902, Paul St\"ackel constructed an analytic function $f(z)$ in a neighborhood of the origin, which was transcendental, and with the property that both $f(z)$ and its inverse, as well as its derivatives, assumed algebraic values at all…

Let $(M_j)_{j=1}^\infty\in\mathbb{N}$ and $(r_j)_{j=1}^\infty\in\mathbb{R}^+$ be increasing sequences satisfying some mild rate of growth conditions. We prove that there is an entire function $f: \mathbb{C} \rightarrow\mathbb{C}$ whose…

We examine the itinerary of $0\in S^{1}=\R/\Z$ under the rotation by $\alpha\in\R\bs\Q$. The motivating question is: if we are given only the itinerary of 0 relative to $I\subset S^{1}$, a finite union of closed intervals, can we recover…

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.…

For a function from the unit interval to itself with constant slope and one discontinuity, the itineraries of the point of discontinuity are called the critical itineraries. These critical itineraries play a significant role in the study of…

Let $f$ be a transcendental entire function. For $n \in \mathbb{N},$ let $ f^{n}$ denote the $n^{th}$ iterate of $f$. Let $ I(f) = \{z \in \mathbb{C} : f^n \rightarrow \infty $ as $ n \rightarrow \infty \} $ and $ K(f) = \{z: \textrm{ there…

We classify transcendental entire functions that are compositions of a polynomial and the exponential for which all singular values escape on disjoint rays. We focus on the case where the escape is degenerate in the sense that points from…

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…

In this paper we present an abstraction-refinement approach to Satisfiability Modulo the theory of transcendental functions, such as exponentiation and trigonometric functions. The transcendental functions are represented as uninterpreted…

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…

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…

In this note we investigate the asymptotic behavior of the number of maximum modulus points, of an entire function, sitting in a disc of radius $r$. In 1964, Erd\Humlaut{o}s asked whether there exists a non-monomial function so that this…

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…

We study the escaping set of functions in the class $\mathcal B^*$, that is, holomorphic functions $f:\mathbb C^*\to\mathbb C^*$ for which both zero and infinity are essential singularities, and the set of singular values of $f$ is…

Previously published admissibility conditions for an element of $\{0,1\}^{\mathbb{Z}}$ to be the itinerary of a point of the inverse limit of a tent map are expressed in terms of forward orbits. We give necessary and sufficient conditions…

Let f be a function transcendental and meromorphic in the plane, and define g(z) by g(z) = f(z+1) - f(z). A number of results are proved concerning the existence of zeros of g(z) or g(z)/f(z), in terms of the growth and the poles of f.

For a transcendental entire function f, we study the set of points BU(f) whose iterates under f neither escape to infinity nor are bounded. We give new results on the connectedness properties of this set and show that, if U is a Fatou…