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 . We prove that if σ₀ ∈ ∂₀, then the Hausdorff dimension of the Julia-Lavaurs set is continuous at σ₀ as the function of the parameter if and only if . Since on a dense set of parameters which correspond to preparabolic points, the lower semicontinuity implies the continuity of on an open and dense subset of ∂₀.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm214-2-2, author = {Ludwik Jaksztas}, title = {On the continuity of the Hausdorff dimension of the Julia-Lavaurs sets}, journal = {Fundamenta Mathematicae}, volume = {215}, year = {2011}, pages = {119-133}, zbl = {1280.37046}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm214-2-2} }
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/