Abstract
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.
Highlights
If f has finite order of growth, 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 every repelling or parabolic periodic point is the landing point of at least one periodic hair
We show that every periodic dreadlock lands at a repelling or parabolic periodic point, and every repelling or parabolic periodic point is the landing point of at least one periodic dreadlock
Often contains curves to infinity; in some cases this was already noticed by Fatou [Fat26]. It was the work of Devaney and his collaborators that really began the study of these hairs or dynamic rays in the 1980s, for functions in the exponential family, fa : z → ez + a
Summary
In their study of the dynamics of complex polynomials and the Mandelbrot set [DH85], Douady and Hubbard introduced the notion of external rays, which can be characterised as the gradient lines of the Green’s function on the basin of attraction of infinity, C\K(p). Our goal is to extend the Douady-Hubbard landing theorem to the case of a transcendental entire function f. In this setting, the role that critical values play in polynomial dynamics is taken by the larger set S(f ) of singular values of f. It was the work of Devaney and his collaborators (see e.g. [DK84, DT86]) that really began the study of these hairs or dynamic rays in the 1980s, for functions in the exponential family, fa : z → ez + a
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