Let $f\colon \widehat{\mathbb{C}}\to \widehat{\mathbb{C}}$ be a rational function of degree two or greater whose Julia set $J_f$ is connected. If $S\subseteq J_f$ is a finite set of periodic points, is it possible that the complement $J_f\setminus S$ has infinitely many connected components? I am particularly interested in the case where $f$ is hyperbolic.