Are there critical points on the boundaries of mother hedgehogs?

Dedicated to the memory of Adrien Douady.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 readswhere 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.

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“…Theorem 4.3 uses notation from Subsection 2.2 and ideas of [Chi06]. It implies Theorem 1.1(1) for connected Julia sets.…”

Section: 2mentioning

confidence: 89%

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

Blokh

Childers

Levin

et al. 2016

Advances in Mathematics

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“…Theorem D follows from the following proposition and Theorem A. The above proposition is proved for the quadratic maps Q α which are not linearizable at 0 in [Chi08], where only the second possibility may arise. We break the proof of the proposition into several lemmas.…”

Section: 3mentioning

confidence: 91%

Statistical properties of quadratic polynomials with a neutral fixed point

Ávila

Cheraghi

2018

J. Eur. Math. Soc.

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We describe the statistical properties of the dynamics of the quadratic polynomials P α (z) = e 2παi z + z 2 on the complex plane, with α of high return times. In particular, we show that these maps are uniquely ergodic on their measure theoretic attractors, and the unique invariant probability is a physical measure describing the statistical behavior of typical orbits in the Julia set. This confirms a conjecture of Perez-Marco on the unique ergodicity of hedgehog dynamics, in this class of maps.2010 Mathematics Subject Classification. Primary: 37F50, Secondary: 37F25, 58F11. The second author acknowledges funding from Leverhulme Trust while carrying out this research. 1The domain U contains 0 and cp P , but not the other critical point of P at −1.Following [IS06], we define the class of mapsand ϕ has a quasi-conformal extension onto C. .Every map in IS has a fixed point of multiplier one at 0, and a unique critical point at cp h = ϕ(−1/3) ∈ U h which is mapped to cv h = −4/27. Elements of IS have the same covering structure as the one of P on U.

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“…One more fact we need to recall is that the hedgehog H does not intersect with CO. (But H is contained in P ¼ CO. See [3].) In fact, if H > CO -Y, the only critical point of f ðzÞ ¼ z 2 þ c eventually captured in H ð¼ f n ðHÞ for all n $ 0Þ.…”

Section: Hedgehogs and Hyperbolic Leaves Of Gross Typementioning

confidence: 99%

“…One can easily extend Theorems 4.2 and 4.3 to the case of Siegel hedgehogs (linearizable hedgehogs). See [9] (V.1) and [3].…”

Section: Journal Of Difference Equations and Applications 709mentioning

confidence: 99%

On the natural extensions of dynamics with a Siegel or Cremer point

Cabrera

Kawahira

2013

Journal of Difference Equations and Applications

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In this note, we show that the regular part of the natural extension (in the sense of Lyubich and Minsky 1997, J. Diff. Geom., 49, pp. 17 -94) of quadratic map f ðzÞ ¼ e 2piu z þ z 2 with irrational u of bounded type has only parabolic leaves except the invariant lift of the Siegel disc. We also show that though the natural extension of a rational function with a Cremer fixed point has a continuum of irregular points, it cannot supply enough singularity to apply the Gross star theorem to find hyperbolic leaves.

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Are there critical points on the boundaries of mother hedgehogs?

Cited by 7 publications

References 0 publications

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

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

Statistical properties of quadratic polynomials with a neutral fixed point

On the natural extensions of dynamics with a Siegel or Cremer point

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