In 1988, Mayer proved the remarkable fact that ∞ is an explosion point
for the set E( fa) of endpoints of the Julia set of fa : C → C; ez + a with a < −1;
that is, the set E( fa) is totally separated (in particular, it does not have any non-trivial
connected subsets), but E( fa)∪{∞} is connected. Answering a question of Schleicher,
we extend this result to the set E ˜( fa) of escaping endpoints in the sense of Schleicher
and Zimmer, for any parameter a ∈ C for which the singular value a 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 E ˜( f ) ∪ {∞} is connected
for any transcendental entire function f of finite order with bounded singular set. We
also discuss corresponding results for all endpoints in the case of exponential maps;
to do so, we establish a version of Thurston’s no wandering triangles theorem for
exponential maps.

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• Escaping Endpoints Explode