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

Ludwik Jaksztas

Ludwik Jaksztas

Fundamenta Mathematicae, Tome 215 (2011), p. 119-133 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $σ \in \bar{₀}$ if and only if $H D (J_{0, σ ₀}) \geq 4 / 3$ . Since $H D (J_{0, σ}) > 4 / 3$ on a dense set of parameters which correspond to preparabolic points, the lower semicontinuity implies the continuity of $H D (J_{0, σ})$ on an open and dense subset of ∂₀.

Publié le : 2011-01-01

Zbl 1280.37046

EUDML-ID : urn:eudml:doc:283059

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     title = {On the continuity of the Hausdorff dimension of the Julia-Lavaurs sets},
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     year = {2011},
     pages = {119-133},
     zbl = {1280.37046},
     language = {en},
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Ludwik Jaksztas. On the continuity of the Hausdorff dimension of the Julia-Lavaurs sets. Fundamenta Mathematicae, Tome 215 (2011) pp. 119-133. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm214-2-2/