Erdős space in Julia sets

David Sumner Lipham

Published 2020-04-27Version 1

Let $\dot E_{\text{Im}}$ be set of endpoints of the Julia set of $\exp(z)-1$ whose forward orbits escape to infinity in the imaginary direction. We prove $\dot E_{\text{Im}}$ is homeomorphic to Erd\H{o}s space $\mathfrak E:=\{x\in \ell^2:x_n\in \mathbb Q\text{ for all }n<\omega\}.$ This result extends to all complex exponential functions $\exp(z)+a$ which have attracting or parabolic periodic orbits.