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

Ludwik Jaksztas¹

¹ Faculty of Mathematics and Information Sciences Warsaw University of Technology Pl. Politechniki 1 00-661 Warszawa, Poland

Fundamenta Mathematicae, Tome 214 (2011) no. 2, pp. 119-133

Résumé

Let $f_0(z)=z^2+1/4$. We denote by $\mathcal{E}_0$ the set of parameters $\sigma\in\mathbb C$ for which the critical point 0 escapes from the filled-in Julia set $K(f_{0})$ in one step by the Lavaurs map $g_\sigma$. We prove that if $\sigma_0\in\partial\mathcal E_0$, then the Hausdorff dimension of the Julia–Lavaurs set $J_{0,\sigma}$ is continuous at $\sigma_0$ as the function of the parameter $\sigma\in\overline{\mathcal E_0}$ if and only if ${\rm HD}(J_{0,\sigma_0})\geq4/3$. Since ${\rm HD}(J_{0,\sigma})>4/3$ on a dense set of parameters which correspond to preparabolic points, the lower semicontinuity implies the continuity of ${\rm HD}(J_{0,\sigma})$ on an open and dense subset of $\partial\mathcal{E}_0$.

Détail

DOI : 10.4064/fm214-2-2

Keywords: denote mathcal set parameters sigma mathbb which critical point escapes filled in julia set step lavaurs map sigma prove sigma partial mathcal hausdorff dimension julia lavaurs set sigma continuous sigma function parameter sigma overline mathcal only sigma geq since sigma dense set parameters which correspond preparabolic points lower semicontinuity implies continuity sigma dense subset partial mathcal