Eremenko points and the structure of the escaping set

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 the fast escaping set. We give analogous results for the intersection of I(f) with the Julia set when multiply connected wandering domains are not present, and show that completely different results hold 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.