Another almost zero-dimensional space of exact multiplicative class 3

David S. Lipham

Published 2020-10-26Version 1

We show that the escaping set for $f(z)=\exp(z)-1$ is nowhere $\sigma$-complete. This establishes that the escaping endpoint set $\dot E(f)$ is a first category almost zero-dimensional space which is $F_{\sigma\delta}$ and nowhere $G_{\delta \sigma}$. Only two other elementary spaces with those properties are known: $\mathbb Q ^\omega$ and Erd\H{o}s space $\mathfrak E$. Previous work has shown that $\dot E(f)$ is homeomorphic to neither of those.