# Cutting a Julia set into infinitely many pieces at finitely many points

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.

Well, $$f(z)=z^2$$ has degree 2, and $$S=\{1,-1\}$$ is a set of periodic points. Then $$J_f \setminus S$$ consists of two components.