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

Alexander Blokh*, Doug Childers, Genadi Levin, Lex Oversteegen, Dierk Schleicher

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

8 Scopus citations

Abstract

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-(pre)critical 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 Ñ+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-(pre)critical branch continua, χ=1 if BC is non-empty, and χ=0 otherwise.

Original languageAmerican English
Pages (from-to)1121-1174
Number of pages54
JournalAdvances in Mathematics
Volume288
DOIs
StatePublished - 22 Jan 2016

Bibliographical note

Publisher Copyright:
© 2015 Elsevier Inc.

Keywords

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

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