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Abstract

Given $d \ge 2$ consider the family of polynomials $P_c(z) = z^d + c$ for $c \in {\mathbb C}$. Denote by $J_c$ the Julia set of $P_c$ and let ${\cal M}_d = \{ c \mid J_c \hbox{ is connected}\}$ be the connectedness locus; for $d = 2$ it is called the Mandelbrot set. We study semihyperbolic parameters $c_0 \in \partial {\cal M}_d$: those for which the critical point $0$ is not recurrent by $P_{c_0}$ and without parabolic cycles. The Hausdorff dimension of $J_c$, denoted by ${\rm HD}(J_c)$, does not depend continuously on $c$ at such $c_0 \in \partial{\cal M}_d$; on the other hand the function $c \mapsto{\rm HD}(J_c)$ is analytic in ${\mathbb C} - {\cal M}_d$. Our first result asserts that there is still some continuity of the Hausdorff dimension if one approaches $c_0$ in a “good” way: there is $C = C(c_0) > 0$ such that for a sequence $c_n \rightarrow c_0$, $$\def\dist{\mathop{\rm dist}} \hbox{if}\quad\dist(c_n, {\cal M}_d) \ge C|c_n - c_0|^{1 + {1}/{d}}, \quad \hbox{then}\quad {\rm HD}(J_{c_n}) \rightarrow {\rm HD} (J_{c_0}). $$ To prove this we use the fact that ${\cal M}_d$ and $J_{c_0}$ are similar near $c_0$. In fact we prove that the biholomorphism $\psi : \overline{\mathbb C} - J_{c_0} \rightarrow \overline{\mathbb C} - {\cal M}_d$ tangent to the identity at infinity is conformal at $c_0$: there is $\lambda \neq 0$ such that $$ \psi(w) = c_0 + \lambda(w - c_0) + {\cal O}(|w - c_0|^{1 + {1}/{d}})\quad\ \hbox{for } w \not \in J_{c_0}. $$ This implies that the local structures of ${\cal M}_d$ and $J_{c_0}$ at $c_0$ are similar. The fact that $\lambda \neq 0$ 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 $C > 0$, $$ d_{\rm H}(J_c, J_{c_0}) \le C|c - c_0|^{{1}/{d}} \quad\hbox{and}\quad d_{\rm H}(K_c, J_{c_0}) \le C|c - c_0|^{{1}/{d}}, $$ where $d_{\rm H}$ denotes the Hausdorff distance.