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Given d ≥ 2 consider the family of polynomials Pc(z) = z + c for c ∈ C. Denote by Jc the Julia set of Pc and letMd = {c | Jc is connected} be the connectedness locus; for d = 2 it is called the Mandelbrot set. We study semihyperbolic parameters c0 ∈ ∂Md: those for which the critical point 0 is not recurrent by Pc0 and without parabolic cycles. The Hausdorff dimension of Jc, denoted by HD(Jc), does not depend continuously on c at such c0 ∈ ∂Md; on the other hand the function c 7→ HD(Jc) is analytic in C − Md. Our first result asserts that there is still some continuity of the Hausdorff dimension if one approaches c0 in a “good” way: there is C = C(c0) > 0 such that for a sequence cn → c0, if dist(cn,Md) ≥ C|cn − c0|, then HD(Jcn)→ HD(Jc0). To prove this we use the fact that Md and Jc0 are similar near c0. In fact we prove that the biholomorphism ψ : C−Jc0 → C−Md tangent to the identity at infinity is conformal at c0: there is λ 6= 0 such that ψ(w) = c0 + λ(w − c0) +O(|w − c0|) for w 6∈ Jc0 . This implies that the local structures ofMd and Jc0 at c0 are similar. The fact that λ 6= 0 is related to a transversality phenomenon that is well known for Misiurewicz parameters and that we extend to the semihyperbolic case. We also prove that for some C > 0, dH(Jc, Jc0) ≤ C|c− c0| and dH(Kc, Jc0) ≤ C|c− c0|, where dH denotes the Hausdorff distance.

On the continuity of Hausdorff dimension of Julia sets and similarity between the Mandelbrot set and Julia sets