The escaping set of the exponential

We show that the points that converge to infinity under iteration of the exponential map form a connected subset of the complex plane.Comment: 5 pages, 1 figur

Non-escaping endpoints do not explode

The family of exponential maps ƒα(z)=ez + α is of fundamental importance in the study of transcendental dynamics. Here we consider the topological structure of certain subsets of the Julia set J(ƒα). When α ∈ (−∞,−1), and more generally when α belongs to the Fatou set F(ƒα), it is known that J(ƒα) can be written as a union of hairs and endpoints of these hairs. In 1990, Mayer proved for α ∈ (−∞,−1) that, while the set of endpoints is totally separated, its union with infinity is a connected set. Recently, Alhabib and the second author extended this result to the case where α ∈ F(ƒα), and showed that it holds even for the smaller set of all escaping endpoints. We show that, in contrast, the set of non-escaping endpoints together with infinity is totally separated. It turns out that this property is closely related to a topological structure known as a ‘spider’s web’; in particular we give a new topological characterisation of spiders’ webs that maybe of independent interest. We also show how our results can be applied to Fatou’s function, z ↦ z + 1 + e−z

Brushing the hairs of transcendental entire functions

Let f be a hyperbolic transcendental entire function of finite order in the Eremenko-Lyubich class (or a finite composition of such maps), and suppose that f has a unique Fatou component. We show that the Julia set of $f$ is a Cantor bouquet; i.e. is ambiently homeomorphic to a straight brush in the sense of Aarts and Oversteegen. In particular, we show that any two such Julia sets are ambiently homeomorphic. We also show that if f\in\B has finite order (or is a finite composition of such maps), but is not necessarily hyperbolic, then the Julia set of f contains a Cantor bouquet. As part of our proof, we describe, for an arbitrary function f\in\B, a natural compactification of the dynamical plane by adding a "circle of addresses" at infinity.Comment: 19 pages. V2: Small number of minor corrections made from V

An answer to a question of Herman, Baker and Rippon concerning Siegel disks

We give a positive answer to a question of Herman, Baker and Rippon by showing that, in the exponential family, the boundary of every unbounded Siegel disk contains the singular value.Comment: 4 pages, 1 figure; appeared as preprint 02/7 in the Berichtsreihe des Mathematischen Seminars der CAU Kiel (2002

Connected escaping sets of exponential maps

We show that for many complex parameters a, the set of points that converge to infinity under iteration of the exponential map f(z)=e^z+a is connected. This includes all parameters for which the singular value escapes to infinity under iteration.Comment: 9 pages; minor revisions from Version

On Nonlanding Dynamic Rays of Exponential Maps

We consider the case of an exponential map for which the singular value is accessible from the set of escaping points. We show that there are dynamic rays of which do not land. In particular, there is no analog of Douady's ``pinched disk model'' for exponential maps whose singular value belongs to the Julia set. We also prove that the boundary of a Siegel disk $U$ for which the singular value is accessible both from the set of escaping points and from $U$ contains uncountably many indecomposable continua.Comment: 15 pages; 1 figure. V2: A result on Siegel disks, as well as a figure, has been added. Some minor corrections were also mad

Topological Dynamics of Exponential Maps on their Escaping Sets

We develop an abstract model for the dynamics of an exponential map $z\mapsto \exp(z)+\kappa$ on its set of escaping points and, as an analog of Boettcher's theorem for polynomials, show that every exponential map is conjugate, on a suitable subset of its set of escaping points, to a restriction of this model dynamics. Furthermore, we show that any two attracting and parabolic exponential maps are conjugate on their sets of escaping points; in fact, we construct an analog of Douady's "pinched disk model" for the Julia sets of these maps. On the other hand, we show that two exponential maps are generally not conjugate on their sets of escaping sets. Using the correspondence with our model, we also answer several questions about escaping endpoints of external rays, such as when a ray is differentiable in such an endpoints or how slowly these endpoints can escape to infinity.Comment: 38 pages, 3 figures. // V3: Several typos fixed; some overall revision; parts of the material in Sections 5, 7 and 11 have been rewritte

On a question of Eremenko concerning escaping components of entire functions

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 question of Eremenko.Comment: 7 pages; 1 figure. V2: Final version (some minor changes