Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6425582 | Advances in Mathematics | 2016 | 54 Pages |
Let P be a polynomial of degree d with Julia set JP. Let NË be the number of non-repelling cycles of P. By the famous Fatou-Shishikura inequality NËâ¤dâ1. 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 QiâJPall of whose images are pairwise disjoint, contain no critical points, and contain the limit sets of eval(Qi)â¥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 readsNË+Nirr+Ï+âi(eval(Qi)â2)â¤dâ1 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, Ï=1 if BC is non-empty, and Ï=0 otherwise.