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

Cantor bouquets, explosions, and Knaster continua : dynamics of complex exponentials

We describe some of the interesting dynamical and topological properties of the complex exponential family [lambda]ez and its associated Julia sets

The exponential map is chaotic: An invitation to transcendental dynamics

We present an elementary and conceptual proof that the complex exponential map is "chaotic" when considered as a dynamical system on the complex plane. (This result was conjectured by Fatou in 1926 and first proved by Misiurewicz 55 years later.) The only background required is a first undergraduate course in complex analysis.Comment: 22 pages, 4 figures. (Provisionally) accepted for publication the American Mathematical Monthly. V2: Final pre-publication version. The article has been revised, corrected and shortened by 14 pages; see Version 1 for a more detailed discussion of further properties of the exponential map and wider transcendental dynamic

Measurable dynamics of meromorphic maps

We study how the orbits of the singularities of the inverse of a meromorphic function prescribe the dynamics on its Julia set, at least up to a set of (Lebesgue) measure zero. We concentrate on a family of entire transcendental functions with only finitely many singularities of the inverse, counting multiplicity, all of which either escape exponentially fast or are pre-periodic. For these functions we are able to decide, whether the function is recurrent or not. In the case that the Julia set is not the entire plane we also obtain estimates for the measure of the Julia set. MSC: 37F10

Rigidity of escaping dynamics for transcendental entire functions

We prove an analog of Boettcher's theorem for transcendental entire functions in the Eremenko-Lyubich class B. More precisely, let f and g be entire functions with bounded sets of singular values and suppose that f and g belong to the same parameter space (i.e., are *quasiconformally equivalent* in the sense of Eremenko and Lyubich). Then f and g are conjugate when restricted to the set of points which remain in some sufficiently small neighborhood of infinity under iteration. Furthermore, this conjugacy extends to a quasiconformal self-map of the plane. We also prove that this conjugacy is essentially unique. In particular, we show that an Eremenko-Lyubich class function f has no invariant line fields on its escaping set. Finally, we show that any two hyperbolic Eremenko-Lyubich class functions f and g which belong to the same parameter space are conjugate on their sets of escaping points.Comment: 28 pages; 2 figures. Final version (October 2008). Various modificiations were made, including the introduction of Proposition 3.6, which was not formally stated previously, and the inclusion of a new figure. No major changes otherwis