The radial Julia set of $\exp(z)-2$ is zero-dimensional

David S. Lipham

Published 2020-02-03Version 1

Let $a\in (-\infty,-1)$, let $f_a$ be the complex exponential mapping $z\mapsto e^z+a$, and let $J(f_a)$ denote the Julia set of $f_a$. We show the radial Julia set $\{z\in J(f_a):f_a^n(z)\not\to\infty\}$ has topological dimension zero. This improves a 2018 result by Vasiliki Evdoridou and Lasse Rempe-Gillen. We put $a=-2$ to see that for Fatou's function $f(z)=z+1+e^{-z}$, the entire non-escaping set $\{z\in \mathbb C:f^n(z)\not\to\infty\}$ is zero-dimensional.