Distinguishing endpoint sets from Erdős space

Abstract

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

11 pages, 4 figures

References in corpus (2)

• The topological dimension of radial Julia sets
• On cohesive almost zero-dimensional spaces