Escaping endpoints explode

TL;DR

This paper extends the understanding of the topological structure of escaping endpoints in Julia sets of exponential maps and other transcendental entire functions, showing their union with infinity is connected under broad conditions.

Contribution

It generalizes Mayer's 1988 result to a wider class of exponential maps and transcendental functions, establishing connectedness of escaping endpoints with infinity.

Findings

Escaping endpoints form a totally separated set with infinity connected to it.

The union of escaping endpoints and infinity is connected for broad classes of functions.

A version of Thurston's 'no wandering triangles' theorem is established for endpoints.

Abstract

In 1988, Mayer proved the remarkable fact that infinity is an explosion point for the set of endpoints of the Julia set of an exponential map that has an attracting fixed point. That is, the set is totally separated (in particular, it does not have any nontrivial connected subsets), but its union with the point at infinity is connected. Answering a question of Schleicher, we extend this result to the set of "escaping endpoints" in the sense of Schleicher and Zimmer, for any exponential map for which the singular value belongs to an attracting or parabolic basin, has a finite orbit, or escapes to infinity under iteration (as well as many other classes of parameters). Furthermore, we extend one direction of the theorem to much greater generality, by proving that the set of escaping endpoints joined with infinity is connected for any transcendental entire function of finite order with bounded singular set. We also discuss corresponding results for *all* endpoints in the case of exponential maps; in order to do so, we establish a version of Thurston's "no wandering triangles" theorem.