On the continuity of the Hausdorff dimension of the Julia-Lavaurs sets

Ludwik Jaksztas. "On the continuity of the Hausdorff dimension of the Julia-Lavaurs sets." Fundamenta Mathematicae 214.2 (2011): 119-133. <http://eudml.org/doc/283059>.

@article{LudwikJaksztas2011,
abstract = {Let f₀(z) = z²+1/4. We denote by ₀ the set of parameters σ ∈ ℂ for which the critical point 0 escapes from the filled-in Julia set K(f₀) in one step by the Lavaurs map $g_σ$. We prove that if σ₀ ∈ ∂₀, then the Hausdorff dimension of the Julia-Lavaurs set $J_\{0,σ\}$ is continuous at σ₀ as the function of the parameter $σ ∈ \overline\{₀\}$ if and only if $HD(J_\{0,σ₀\}) ≥ 4/3$. Since $HD(J_\{0,σ\}) > 4/3$ on a dense set of parameters which correspond to preparabolic points, the lower semicontinuity implies the continuity of $HD(J_\{0,σ\})$ on an open and dense subset of ∂₀.},
author = {Ludwik Jaksztas},
journal = {Fundamenta Mathematicae},
keywords = {Hausdorff dimension; Julia-Lavaurs set; continuity; preparabolic points; lower bound},
language = {eng},
number = {2},
pages = {119-133},
title = {On the continuity of the Hausdorff dimension of the Julia-Lavaurs sets},
url = {http://eudml.org/doc/283059},
volume = {214},
year = {2011},
}

TY - JOUR
AU - Ludwik Jaksztas
TI - On the continuity of the Hausdorff dimension of the Julia-Lavaurs sets
JO - Fundamenta Mathematicae
PY - 2011
VL - 214
IS - 2
SP - 119
EP - 133
AB - Let f₀(z) = z²+1/4. We denote by ₀ the set of parameters σ ∈ ℂ for which the critical point 0 escapes from the filled-in Julia set K(f₀) in one step by the Lavaurs map $g_σ$. We prove that if σ₀ ∈ ∂₀, then the Hausdorff dimension of the Julia-Lavaurs set $J_{0,σ}$ is continuous at σ₀ as the function of the parameter $σ ∈ \overline{₀}$ if and only if $HD(J_{0,σ₀}) ≥ 4/3$. Since $HD(J_{0,σ}) > 4/3$ on a dense set of parameters which correspond to preparabolic points, the lower semicontinuity implies the continuity of $HD(J_{0,σ})$ on an open and dense subset of ∂₀.
LA - eng
KW - Hausdorff dimension; Julia-Lavaurs set; continuity; preparabolic points; lower bound
UR - http://eudml.org/doc/283059
ER -