TITLE: Cutting a Julia set into infinitely many pieces at finitely many points
QUESTION [6 upvotes]: 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.

REPLY [9 votes]: For a polynomial, this is equivalent to asking whether there can be infinitely many external rays landing at a single point. This could happen only if the function has a Cremer point (i.e., a non-linearisable irrationally indifferent periodic point). If the Cremer point is accessible from the complement, then any external ray landing at it would have to be non-periodic by the so-called "snail lemma", which means that then there would indeed be infinitely many rays landing at the same point. However, I believe that it is still an open question whether or not this actually happens.
Since hyperbolic maps have no Cremer points, what you ask about is impossible in the hyperbolic case. I think that, similarly, for hyperbolic rational maps it is impossible, and probably more generally whenever the Julia set is locally connected.