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2001 | 170 | 3 | 287-317
Tytuł artykułu

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

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Given d ≥ 2 consider the family of polynomials $P_c(z) = z^d + c$ for c ∈ ℂ. Denote by $J_c$ the Julia set of $P_c$ and let $ℳ_{d} = {c | J_c is connected}$ be the connectedness locus; for d = 2 it is called the Mandelbrot set. We study semihyperbolic parameters $c₀ ∈ ∂ℳ_{d}$: those for which the critical point 0 is not recurrent by $P_{c₀}$ and without parabolic cycles. The Hausdorff dimension of $J_c$, denoted by $HD(J_c)$, does not depend continuously on c at such $c₀ ∈ ∂ℳ_{d}$; on the other hand the function $c ↦ HD(J_c)$ is analytic in $ℂ - ℳ_{d}$. 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 $dist(cₙ,ℳ_{d}) ≥ C|cₙ - c₀|^{1+1/d}$, then $HD(J_{cₙ}) → HD(J_{c₀})$.
To prove this we use the fact that $ℳ_{d}$ and $J_{c₀}$ are similar near c₀. In fact we prove that the biholomorphism $ψ : ℂ̅ - J_{c₀} → ℂ̅ - ℳ_{d}$ tangent to the identity at infinity is conformal at c₀: there is λ ≠ 0 such that
$ψ(w) = c₀ + λ(w-c₀) + 𝒪(|w - c₀|^{1+1/d})$ for $w ∉ J_{c₀}$.
This implies that the local structures of $ℳ_{d}$ and $J_{c₀}$ at c₀ are similar. The fact that λ ≠ 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,
$d_{H}(J_c,J_{c₀}) ≤ C|c-c₀|^{1/d}$ and $d_{H}(K_c,J_{c₀}) ≤ C|c-c₀|^{1/d}$,
where $d_{H}$ denotes the Hausdorff distance.
Słowa kluczowe
  • Institute for Mathematical Sciences, SUNY at Stony Brook, Stony Brook, NY 11794-3660, U.S.A.
  • Universidad Católica del Norte, Avenida Angamos 0610, Antofagasta, Chile
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