The Fatou inequality and wandering continua
by
Lex Oversteegen
UAB
Coauthors: A. Blokh, D. Childers, G. Levin and D. Schleicher

Let P be a polynomial of degree d with Julia set J_P. Let N be the number of cycles of bounded Fatou domains of P plus the number of Cremer periodic orbits of P. Then the number of attracting and neutral periodic cycles of P is less than or equal to N. The famous Fatou-Douady-Hubbard-Shishikura inequality states that N ≤ d-1. The main goal of this paper is to improve this bound.

Denote by N_∞ the number of repelling periodic orbits at which infinitely many external rays land (such orbits do not exist if J_P is connected). A wandering collection is a collection of continua/points W ⊂ J_P with pairwise disjoint images. It is known that wandering collections exist.

We show that if W ⊂ E, where E is a component of J_P, and E\W is empty or disconnected, then there is a finite set A(W) ≠ \0 of external rays with principal sets in W. Call W non-precritical if sⁿ_d|_A(W) is one-to-one for any n (s_d=z^d|_S¹: S¹→ S¹ is the map acting on the angles in the circle at infinity). Given a non-empty wandering collection G of non-precritical continua/points Q₁, ..., Q_k in J_P with |A(Q_i)| > 2 (1 ≤ i ≤ k), we prove that

å G  (|A(Qi)|-2)+N+N∞ ≤ d-2.

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Date received: February 18, 2009