# 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

top

## Abstract

top
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 ${g}_{\sigma }$. We prove that if σ₀ ∈ ∂₀, then the Hausdorff dimension of the Julia-Lavaurs set ${J}_{0,\sigma }$ is continuous at σ₀ as the function of the parameter $\sigma \in \overline{₀}$ if and only if $HD\left({J}_{0,\sigma ₀}\right)\ge 4/3$. Since $HD\left({J}_{0,\sigma }\right)>4/3$ on a dense set of parameters which correspond to preparabolic points, the lower semicontinuity implies the continuity of $HD\left({J}_{0,\sigma }\right)$ on an open and dense subset of ∂₀.

## How to cite

top

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 -

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.