## Abstract

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.

Original language | American English |
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Pages (from-to) | 1121-1174 |

Number of pages | 54 |

Journal | Advances in Mathematics |

Volume | 288 |

DOIs | |

State | Published - 22 Jan 2016 |

### Bibliographical note

Publisher Copyright:© 2015 Elsevier Inc.

## Keywords

- Complex dynamics
- Julia set
- Primary
- Secondary
- Wandering continuum