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February 2013 Continuity of Julia sets and its Hausdorff dimension of Pc(z) = zd + c
Wei ZHUANG
Hokkaido Math. J. 42(3): 385-395 (February 2013). DOI: 10.14492/hokmj/1384273388

Abstract

Given d ≥ 2 consider the family of monic polynomials Pc(z) = zd + c, for c ∈ ℂ. Denote by Jc and HD(Jc) the Julia set of Pc and the Hausdorff dimension of Jc respectively, and let $\mathcal{M}$d = {c|Jc is connected} be the connectedness locus; for d = 2 it is called the Mandelbrot set. We study semihyperbolic parameters c0 ∈ ∂ $\mathcal{M}$d: those for which the critical point is not recurrent by Pc0, 0 ∈ Jc0, and without parabolic cycles. We prove that if PcnPc0 algebraically, then for some C > 0, dH(Jcn, Jc0) ≤ C|cn - c0|1/d, where dH denotes the Hausdorff distance. If, in addition, PcnPc0 preserving critical relations, then Pcn is semihyperbolic for all n ≫ 0, and HD(Jcn) Ⅺ HD(Jc0).

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Wei ZHUANG. "Continuity of Julia sets and its Hausdorff dimension of Pc(z) = zd + c." Hokkaido Math. J. 42 (3) 385 - 395, February 2013. https://doi.org/10.14492/hokmj/1384273388

Information

Published: February 2013
First available in Project Euclid: 12 November 2013

zbMATH: 1377.37073
MathSciNet: MR3137391
Digital Object Identifier: 10.14492/hokmj/1384273388

Subjects:
Primary: 37F35

Rights: Copyright © 2013 Hokkaido University, Department of Mathematics

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Vol.42 • No. 3 • February 2013
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