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

Juan Rivera-Letelier. "On the continuity of Hausdorff dimension of Julia sets and similarity between the Mandelbrot set and Julia sets." Fundamenta Mathematicae 170.3 (2001): 287-317. <http://eudml.org/doc/283213>.

@article{JuanRivera2001,
abstract = {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 $ψ(w) = c₀ + λ(w-c₀) + (|w - c₀|^\{1+1/d\})$ 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.},
author = {Juan Rivera-Letelier},
journal = {Fundamenta Mathematicae},
keywords = {Mandelbrot set; Hausdorff dimension; continuity},
language = {eng},
number = {3},
pages = {287-317},
title = {On the continuity of Hausdorff dimension of Julia sets and similarity between the Mandelbrot set and Julia sets},
url = {http://eudml.org/doc/283213},
volume = {170},
year = {2001},
}

TY - JOUR
AU - Juan Rivera-Letelier
TI - On the continuity of Hausdorff dimension of Julia sets and similarity between the Mandelbrot set and Julia sets
JO - Fundamenta Mathematicae
PY - 2001
VL - 170
IS - 3
SP - 287
EP - 317
AB - 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 $ψ(w) = c₀ + λ(w-c₀) + (|w - c₀|^{1+1/d})$ 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.
LA - eng
KW - Mandelbrot set; Hausdorff dimension; continuity
UR - http://eudml.org/doc/283213
ER -