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

Let P be a polynomial of degree d with Julia set J_P. 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 Q_i⊂J_P all of whose images are pairwise disjoint, contain no critical points, and contain the limit sets of eval(Q_i)≥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 reads Ñ+N_irr + χ + ∑_i(eval(Q_i)-2) ≤ d - 1 where N_irr counts repelling cycles with no periodic rays landing at points in the cycle, {Q_i} form a wandering collection B_C of non-(pre)critical branch continua, χ=1 if BC is non-empty, and χ=0 otherwise.