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

Volume 214 / 2011

Ludwik Jaksztas Fundamenta Mathematicae 214 (2011), 119-133 MSC: Primary 37F45; Secondary 37F35. DOI: 10.4064/fm214-2-2


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 $\partial\mathcal{E}_0$.


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

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