The Topological Dimension of Radial Julia Sets

Abstract

We prove that the meandering set for $$f_a(z)=e^z+a$$ is homeomorphic to the space of irrational numbers whenever a belongs to the Fatou set of $$f_a$$ . This extends recent results by Vasiliki Evdoridou and Lasse Rempe. It implies that the radial Julia set of $$f_a$$ has topological dimension zero for all attracting and parabolic parameters, including all $$a\in (-\infty ,-1]$$ . Similar results are obtained for Fatou’s function $$f(z)=z+1+e^{-z}$$ .