The no lakes of Wada theorem in complex dynamics and a new entry in Sullivan's dictionary

Yûsuke Okuyama

Published 2018-06-19Version 1

We show that the Julia set $J(f)$ of a rational function $f$ on $\mathbb{P}^1$ of degree $>1$ is never the boundary of lakes of Wada. Simultaneously, we also show that with respect to the equilibrium measure $\mu_f$ of $f$, the residual Julia set $J_0(f)$ of $f$ is an either full or null set, and $\mu_f(J_0(f))=0$ if and only if there is a Fatou component of $f$ which is totally invariant under $f^2$, and then $J_0(f)=\emptyset$. This in particular yields the dynamical counterpart to Abikoff's theorem in the theory of Kleinian groups.