Distinguishing endpoint sets from Erdős space

Abstract

Let Ė(f ) denote the set of all endpoints of the Julia set of f (z) = exp(z) − 1 which escape to infinity under iteration of f . We prove Ė(f ) is not homeomorphic to Q × X for any topological space X. In particular, it is not homeomorphic to Erdős space E. Combined with the already known fact that Ė(f ) is not homeomorphic to complete Erdős space Ec, we conclude that Ė(f ) is a fundamentally new type of endpoint set in complex dynamics. Our proof shows that the set of all points z ∈ C whose orbits either escape to ∞ or attract to 0 is path-connected. This result may be of independent interest.