A landing theorem for entire functions with bounded post-singular sets

The emphDouady-Hubbard landing theorem for periodic external rays is one of the cornerstones of the study of polynomial dynamics. It states that, for a complex polynomial $f$ with bounded postcritical set, every periodic external ray lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic point is the landing point of at least one periodic external ray. We prove an analogue of this theorem for an entire function $f$ with bounded postsingular set: every periodic dreadlock lands at a repelling or parabolic periodic point, and conversely every repelling or periodic parabolic point is the landing point of at least one periodic dreadlock. (Here, emphdreadlocks are certain connected subsets of the escaping set of $f$). If, in addition, $f$ has finite order of growth, then the dreadlocks are in fact hairs (curves to infinity). More generally, we prove that every point of a hyperbolic set $K$ of $f$ is the landing point of a dreadlock.