A landing theorem for entire functions with bounded post-singular sets
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GAFA Geometric And Functional Analysis
A LANDING THEOREM FOR ENTIRE FUNCTIONS WITH BOUNDED POST-SINGULAR SETS Anna Miriam Benini and
Lasse Rempe
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.
1 Introduction Let p : C → C be a polynomial. The filled-in Julia set K(p) consists of those points z ∈ C whose orbits remain bounded under repeated application of p. 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). Periodic (and pre-periodic) rays are of particular importance, due to the following result. Anna Miriam Benini: was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 703269 COTRADY and by the SIR Grant NEWHOLITE No. RBSI14CFME. The second author was partially supported by a Philip Leverhulme Prize Keywords and phrases: Transcendental entire function, Transcendental dynamics, Accessibility, Combinatorics, External ray, Hair, Dreadlock Mathematics Subject Classification: Primary 37F20; Secondary 30D05, 37F10, 37F12
1466
A. M. BENINI, L. REMPE
GAFA
Douady-Hubbard landing theorem. Let p be a polynomial whose post-critical set {pn (c) : n ≥ 1} (1.1) P(p) ..= c : p (c)=0
is bounded. (Equivalently, assume that K(p) is connected.) Then every periodic ray of p lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point of p is the landing point of at least one and at most finitely many periodic external rays. The first half of this theorem, concerning the landing of periodic rays, can be found in [DH85, Expos´e VIII.II, Proposition 2]. The seco
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