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A landing theorem for entire functions with bounded post-singular sets
Ist Teil von
Geometric and functional analysis, 2020-12, Vol.30 (6), p.1465-1530
Ort / Verlag
Cham: Springer International Publishing
Erscheinungsjahr
2020
Link zum Volltext
Quelle
SpringerLink (Online service)
Beschreibungen/Notizen
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
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 this theorem for an entire function
f
with bounded postsingular set. If
f
has finite order of growth, then it is known that the escaping set
I
(
f
) 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
f
of infinite order, such hairs may not exist. Therefore we introduce certain dynamically natural connected subsets of
I
(
f
), called
dreadlocks
. We show that every periodic dreadlock 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 dreadlock. More generally, we prove that every point of a hyperbolic set is the landing point of a dreadlock.