Hokkaido Mathematical Journal


Continuity of Julia sets and its Hausdorff dimension of Pc(z) = zd + c.

Hokkaido Mathematical Journal, 42 (2013) pp.385-395




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 Pcn Ⅺ Pc0 algebraically, then for some C > 0, dH(Jcn, Jc0) ≤ C|cn - c0|1/d, where dH denotes the Hausdorff distance. If, in addition, Pcn Ⅺ Pc0 preserving critical relations, then Pcn is semihyperbolic for all n ≫ 0, and HD(Jcn) Ⅺ HD(Jc0).

Uncontrolled KeywordsJulia set, Hausdorff dimension, net, conformal measure