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

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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-precritical branch continua, i.e., collections of continua or points Qᵢ⊂ Jₚ all 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 reads N+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-precritical branch continua, χ=1 if BC is non-empty, and χ=0 otherwise.