Distinguishing endpoint sets from Erdős space

Abstract We prove that the set of all endpoints Julia $f(z)=\exp\!(z)-1$ which escape to infinity under iteration f is not homeomorphic rational Hilbert space $\mathfrak E$ . As a corollary, we show points $z\in \mathbb C$ whose orbits either $\infty$ or attract 0 path-connected. extend these results many other functions in exponential family.