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 $\widetilde N$ be the number of non-repelling cycles of $P$. By the famous Fatou-Shishikura inequality $\widetilde N\le d-1$. The goal of the paper is to improve this bound. The new count includes \emph{wandering collections of non-precritical branch continua}, i.e., collections of continua or points $Q_i\subset J_P$ \emph{all} of whose images are pairwise disjoint, contain no critical points, and contain the limit sets of $\mathrm{eval}(Q_i)\ge 3$ 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 \[ \widetilde N+N_{irr}+\chi+\sum_i (\mathrm{eval}(Q_i)-2) \le d-1 \] where $N_{irr}$ counts repelling cycles with no periodic rays landing at points in the cycle, $\{Q_i\}$ form a wandering collection $\mathcal{B}_\mathbb{C}$ of non-precritical branch continua, $\chi=1$ if $\mathcal{B}_\mathbb{C}$ is non-empty, and $\chi=0$ otherwise.