Eremenko points and the structure of the escaping set

Rippon, Philip and Stallard, Gwyneth (2019). Eremenko points and the structure of the escaping set. Transactions of the American Mathematical Society, 372(5) pp. 3083–3111.

DOI: https://doi.org/10.1090/tran/7673

Abstract

Much recent work on the iterates of a transcendental entire function f has been motivated by Eremenko's conjecture that all the components of the escaping set I(f) are unbounded. We prove several general results about the topological structure of I(f) including the fact that if I(f) is disconnected, then it contains uncountably many pairwise disjoint unbounded continua, all of which are subsets of A_R(f), the 'core' of the fast escaping set. We also show that, for some R > 0, the set A_R(f) is connected and has the structure of an infinite spider's web or it contains uncountably many unbounded connected F_σ sets. There are analogous results for the intersections of these sets with the Julia set when multiply connected wandering domains are not present, but very different results when such wandering domains are present. In proving these, we obtain the unexpected result that some types of multiply connected wandering domains have complementary components with no interior, indeed uncountably many.

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