  The Douady-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 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 with bounded
postsingular set. If the function has finite order of growth, then it is known
that the escaping set contains certain curves called "periodic hairs"; we show
that every periodic hair lands at a repelling or parabolic periodic point, and
conversely every repelling or parabolic periodic point is the landing point of
at least one periodic hair. For a postsingularly bounded entire function of
infinite order, such hairs may not exist. Therefore we introduce certain
dynamically natural connected sets, called "filaments". We show that every
periodic filament lands at a repelling or parabolic periodic point, and
conversely every repelling or parabolic periodic point is the landing point of
at least one periodic filament.
  More generally, we prove that every point of a hyperbolic set is the landing
point of a filament.
