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

Fundamenta Mathematicae (2011)

- Volume: 214, Issue: 2, page 119-133
- ISSN: 0016-2736

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topLudwik 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 -

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