# A landing theorem for hairs and dreadlocks for transcendental entire functions

2017/11/10 Fri 14:00 - 17:00

6号館609号室

Anna Miriam Benini

University of Rome Tor Vergata

In polynomial dynamics, the set of escaping points consists of curves equipped with symbolic dynamics, called external rays. 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 $f$ has finite order of growth, then our result again states that every periodic hair of $f$ lands at a repelling or parabolic point, and conversely every repelling or parabolic point is the landing point of at least one periodic hair. (Here a periodic hair is a maximal 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 in our landing theorem is taken by more general connected sets of escaping points, which we call dreadlocks. In fact, we prove more generally that, given any hyperbolic set $K$ of $f$, every point of $K$ is the landing point of a dreadlock.