TY - JOUR

T1 - An extended Fatou-Shishikura inequality and wandering branch continua for polynomials

AU - Blokh, Alexander

AU - Childers, Doug

AU - Levin, Genadi

AU - Oversteegen, Lex

AU - Schleicher, Dierk

N1 - Publisher Copyright:
© 2015 Elsevier Inc.

PY - 2016/1/22

Y1 - 2016/1/22

N2 - 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.

AB - 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.

KW - Complex dynamics

KW - Julia set

KW - Primary

KW - Secondary

KW - Wandering continuum

UR - http://www.scopus.com/inward/record.url?scp=84947770270&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2015.10.020

DO - 10.1016/j.aim.2015.10.020

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AN - SCOPUS:84947770270

SN - 0001-8708

VL - 288

SP - 1121

EP - 1174

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -