An extended Fatou–Shishikura inequality and wandering branch continua for polynomials

Abstract

Let P be a polynomial of degree d with Julia set JP. Let N˜ be the number of non-repelling cycles of P. By the famous Fatou–Shishikura inequality N˜≤d−1. The goal of the paper is to improve this bound. The new count includes wandering collections of non-(pre)critical branch continua, i.e., collections of continua or points Qi⊂JPall of whose images are pairwise disjoint, contain no critical points, and contain the limit sets of eval(Qi)≥3 external rays. Also, we relate individual cycles, which are either non-repelling or repelling with no periodic rays landing, to individual critical points that are recurrent in a weak sense.A weak version of the inequality readsN˜+Nirr+χ+∑i(eval(Qi)−2)≤d−1 where Nirr counts repelling cycles with no periodic rays landing at points in the cycle, {Qi} form a wandering collection BC of non-(pre)critical branch continua, χ=1 if BC is non-empty, and χ=0 otherwise.