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

Juan Rivera-Letelier¹

¹ Institute for Mathematical Sciences SUNY at Stony Brook Stony Brook, NY 11794-3660, U.S.A. and Universidad Católica del Norte Avenida Angamos 0610 Antofagasta, Chile

Fundamenta Mathematicae, Tome 170 (2001) no. 3, pp. 287-317.

Résumé

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.

Détail

DOI : 10.4064/fm170-3-6

Keywords: given consider family polynomials mathbb denote julia set cal mid hbox connected connectedness locus called mandelbrot set study semihyperbolic parameters partial cal those which critical point recurrent without parabolic cycles hausdorff dimension denoted does depend continuously partial cal other function mapsto analytic mathbb cal first result asserts there still continuity hausdorff dimension approaches there sequence rightarrow def dist mathop dist hbox quad dist cal quad hbox quad rightarrow prove cal similar near prove biholomorphism psi overline mathbb rightarrow overline mathbb cal tangent identity infinity conformal there lambda neq psi lambda cal quad hbox implies local structures cal similar lambda neq related transversality phenomenon known misiurewicz parameters extend semihyperbolic prove quad hbox quad where denotes hausdorff distance