Anna Miriam Benini

The Douady-Hubbard landing theorem for periodic external
rays is one of the cornerstones of the successful 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 the theorem for entire functions with
bounded postsingular set. If such $f$ additionally has finite order
of growth, then our result states precisely that every periodic hair
of $f$ lands at a repelling or parabolic point, and again conversely
every repelling or parabolic point is the landing point of at least
one periodic hair. (Here a \emph{periodic hair} is a curve consisting
of escaping points of $f$ that is invariant under an iterate of
$f$.) For general $f$ with bounded postsingular set, but not
necessarily of finite order, the role of hairs is taken by more
general connected sets of escaping points, which we call
\emph{dreadlocks}. This is joint work with Lasse Rempe-Gillen.