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.

Topological Dynamics of Exponential Maps on their Escaping Sets

For a transcendental entire function ƒ, we study the set of points BU (ƒ) whose iterates under ƒ 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 component that meets BU (ƒ), then most boundary points of U (in the sense of harmonic measure) lie in BU (ƒ). We prove this using a new result concerning the set of limit points of the iterates of ƒ on the boundary of a wandering domain. Finally, we give some examples to illustrate our results.

/pdf/on-the-set-where-the-iterates-of-an-entire-function-are-lf0rsbtvu6.pdf

On the set where the iterates of an entire function are neither escaping nor bounded

We consider the case of an exponential map E k : z → exp(z) + k for which the singular value k is accessible from the set of escaping points of E k . We show that there are dynamic rays of E k 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.

/pdf/on-nonlanding-dynamic-rays-of-exponential-maps-4bz22f9jyb.pdf

On nonlanding dynamic rays of exponential maps

/pdf/indecomposable-continua-in-exponential-dynamics-3jal43ku9k.pdf

Indecomposable continua in exponential dynamics

In this paper we discuss the topology and dynamics of Eλ(z) = λez when λ is real and λ > 1/e. It is known that the Julia set of Eλ is the entire plane in this case. Our goal is to show that there are certain natural invariant subsets for Eλ which are topologically Knaster-like continua. Moreover, the dynamical behavior on these invariant sets is quite tame. We show that the only trivial kinds of α- and ω-limit sets are possible.

Stephen B. Curry

Papers

One-dimensional nonseparating plane continua with disjoint ε-dense subcontinua