Buried points and lakes of Wada Continua

Abstract

Let $f:\hat \mathbf C\rightarrow \hat \mathbf C$ be a rational map of degree $n\geq 3$ and with exactly two critical points. Assume that the Julia set $J(f)$ is a proper subcontinuum of $\hat \mathbf C$ and there is no completely invariant Fatou component under the iterates $f^{2}$. It is shown that if there is no buried points in $J(f)$, then the Julia set $J(f)$ is a Lakes of Wada continuum, and hence is either an indecomposable continuum or the union of two indecomposable continua.