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

• Autorzy: Juan Rivera-Letelier
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 37F45: Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations
• 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2001
• Tom: 170
• Numer: 3
• Strony: 287-317

Daty

wydano

2001

Twórcy

autor

Juan Rivera-Letelier

• 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

Identyfikatory

DOI

10.4064/fm170-3-6