External rays to periodic points

Abstract

We prove that for every polynomial-like holomorphic mapP, ifaeK (filled-in Julia set) and the componentK aofK containinga is either a point ora is accessible along a continuous curve from the complement ofK andK ais eventually periodic, thena is accessible along an external ray. Ifa is a repelling or parabolic periodic point, then the set of arguments of the external rays converging toa is a nonempty closed “rotation set”, finite (ifK ais not a one point) or Cantor minimal containing a pair of arguments of external rays of a critical point in ℂ. In the Appendix we discuss constructions via cutting and glueing, fromP to its external map with a “hedgehog”, and backward.