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

Abstract

Given d ≥ 2 consider the family of monic polynomials P_c(z) = z^d + c, for c ∈ ℂ. Denote by J_c and HD(J_c) the Julia set of P_c and the Hausdorff dimension of J_c respectively, and let $\mathcal{M}$_d = {c|J_c 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 P_c0, 0 ∈ J_c0, and without parabolic cycles. We prove that if P_cn Ⅺ P_c0 algebraically, then for some C > 0, d_H(J_cn, J_c0) ≤ C|c_n - c₀|^1/d, where d_H denotes the Hausdorff distance. If, in addition, P_cn Ⅺ P_c0 preserving critical relations, then P_cn is semihyperbolic for all n ≫ 0, and HD(J_cn) Ⅺ HD(J_c0).