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

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

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 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

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 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

Prime Ends and Local Connectivity

Let U be a simply connected domain on the Riemann sphere whose complement K contains more than one point. We establish a characterization of local connectivity of K at a point in terms of the prime ends whose impressions contain this point. Invoking a result of Ursell and Young, we obtain an alternative proof of a theorem of Torhorst, which states that the impression of a prime end of $U$ contains at most two points at which $K$ is locally connected.Comment: 13 pages, 2 figures. V6: Added additional remarks on the mathematical work of Marie Torhorst. Summary of previous versions: V5 - Updated the article by adding a historical note. V4 - Final preprint prior to publication. V3 - Proof of the Ursell-Young theorem and illustrations added. V2 - The original short note was expanded to a full article. V1 - original short not

Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity

We study the bifurcation loci of quadratic (and unicritical) polynomials and exponential maps. We outline a proof that the exponential bifurcation locus is connected; this is an analog to Douady and Hubbard's celebrated theorem that (the boundary of) the Mandelbrot set is connected. For these parameter spaces, a fundamental conjecture is that hyperbolic dynamics is dense. For quadratic polynomials, this would follow from the famous stronger conjecture that the bifurcation locus (or equivalently the Mandelbrot set) is locally connected. It turns out that a formally slightly weaker statement is sufficient, namely that every point in the bifurcation locus is the landing point of a parameter ray. For exponential maps, the bifurcation locus is not locally connected. We describe a different conjecture (triviality of fibers) which naturally generalizes the role that local connectivity has for quadratic or unicritical polynomials.Comment: 20 pages, 6 figure

ESCAPING SETS ARE NOT SIGMA-COMPACT

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.Comment: 6 pages. To appear in Proc. Amer. Math. Soc. V4: Author accepted manuscript. Clarification of an imprecise statement regarding nowhere density in Corollary 2.2 that was present in v