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

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Rivera-Letelier J.

Fundamenta Mathematicae

2001

tom 170

nr 3

287-317

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.

Ograniczanie wyników

1 Fundamenta Mathematicae

1 Rivera-Letelier J.

1 2001

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