External rays to periodic points

We prove that for every polynomial-like holomorphic map P, if a ∈ K (filled-in Julia set) and the component K_a of K containing a is either a point or a is accessible along a continuous curve from the complement of K and K_a is eventually periodic, then a is accessible along an external ray. If a is a repelling or parabolic periodic point, then the set of arguments of the external rays converging to a is a nonempty closed "rotation set", finite (if K_a is not a one point) or Cantor minimal containing a pair of arguments of external rays of a critical point in ℂ\K. In the Appendix we discuss constructions via cutting and glueing, from P to its external map with a "hedgehog", and backward.