 Commuting rational maps of degree at least two have the same Julia set

Let $R,S$ commute. On an open set where $\{R^n\}$ is equicontinuous, so is $\{S\circ R^n\}$, by the rational map is Lipschitz in the chordal metric bound. Since this equals $\{R^n\circ S\}$, equicontinuity descends through the open map $S$: near $S(x)$ every nearby value has a preimage near $x$, and the corresponding estimates hold for all $n$. Thus $S(F(R))\subseteq F(R)$. A degree-at-least-two Julia set contains at least three points, all omitted by the iterates of $S$ on $F(R)$. The three-value Montel theorem gives $F(R)\subseteq F(S)$. Exchanging the maps proves equality of their Fatou and Julia sets. The degree hypothesis excludes Möbius examples with only one Julia point.