Related papers: Escaping endpoints explode

We use Bowen's definition of topological entropy and Ahlfors five islands theorem, as well as the theory of polynomial-like mappings, to show that the topological entropy of any entire transcendental function is infinity. In addition the…

We investigate the set I(f) of points that converge to infinity under iteration of the map f(z) = e^z-1 and show that it is the disjoint union of countably many rays and uncountable union of infinite sets whose points escape to infinity…

We study the dynamics of the exponential maps $E_{\lambda}: \mathbb{C} \longrightarrow \mathbb{C}$ defined by $E_{\lambda}(z) = \lambda e^z$, where $\lambda > \frac{1}{e}$. We prove that for itineraries of a certain form, the set of all…

We construct an entire function in the Eremenko-Lyubich class $\B$ whose Julia set has only bounded path-components. This answers a question of Eremenko from 1989 in the negative. On the other hand, we show that for many functions in $\B$,…

We answer a question of Schleicher by showing that, for an exponential map with nonescaping singular value, every periodic ray lands. This is an analog of a theorem of Douady and Hubbard concerning polynomials. We also prove a partial…

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…

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.…

According to the Thurston No Wandering Triangle Theorem, a branching point in a locally connected quadratic Julia set is either preperiodic or precritical. Blokh and Oversteegen proved that this theorem does not hold for higher degree Julia…

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…

A hyperbolic transcendental entire function with connected Fatou set is said to be "of disjoint type". It is known that a disjoint-type function provides a model for the dynamics near infinity of all maps in the same parameter space; hence…

We give a complete classification of the set of parameters $\kappa$ for which the singular value of $E_{\kappa}:z\mapsto \exp(z)+\kappa$ escapes to infinity under iteration. In particular, we show that every path-connected component of this…

We present proofs of basic results, including those developed by Harold Bell, for the plane fixed point problem: does every map of a non-separating plane continuum have a fixed point? Some of these results had been announced much earlier by…

We study the dynamics of iterated cosine maps $E\colon z \mapsto ae^z+be^{-z},$ with $a,b \in \C\setminus \{0\}$. We show that the points which converge to infinity under iteration are organized in the form of rays and, as in the…

It is known that, for many transcendental entire functions in the Eremenko-Lyubich class $\mathcal{B}$, every escaping point can eventually be connected to infinity by a curve of escaping points. When this is the case, we say that the…

Given a rational map of the Riemann sphere and a subset $A$ of its Julia set, we study the $A$-exceptional set, that is, the set of points whose orbit does not accumulate at $A$. We prove that if the topological entropy of $A$ is less than…

We consider the dynamics of transcendental self-maps of the punctured plane, $\mathbb{C}^*=\mathbb{C}\setminus \{0\}$. We prove that the escaping set $I(f)$ is either connected, or has infinitely many components. We also show that $I(f)\cup…

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 construct exponential maps for which the singular value tends to infinity under iterates while the maps are ergodic. This is in contrast with a result of Lyubich from 1987 which tells that $e^z$ is not ergodic.

Consider the entire function $f(z)=\cosh(z)$. We show that the escaping set of this function - that is, the set of points whose orbits tend to infinity under iteration - has a structure known as a "spider's web". This disproves a conjecture…

Let $f$ be Fatou's function, that is, $f(z)= z+1+e^{-z}$. We prove that the escaping set of $f$ has the structure of a `spider's web' and we show that this result implies that the non-escaping endpoints of the Julia set of $f$ together with…