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- W2964022145 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 N ˜ + 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 B C is non-empty, and χ = 0 otherwise." @default.
- W2964022145 created "2019-07-30" @default.
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- W2964022145 date "2016-01-01" @default.
- W2964022145 modified "2023-10-13" @default.
- W2964022145 title "An extended Fatou–Shishikura inequality and wandering branch continua for polynomials" @default.
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- W2964022145 doi "https://doi.org/10.1016/j.aim.2015.10.020" @default.
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