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

• Autorzy: Ludwik Jaksztas
• Języki publikacji: EN

Abstrakty

EN

Let f₀(z) = z²+1/4. We denote by 𝓔₀ 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 ∈ ∂𝓔₀, then the Hausdorff dimension of the Julia-Lavaurs set $J_{0,σ}$ is continuous at σ₀ as the function of the parameter $σ ∈ \overline{𝓔₀}$ 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 ∂𝓔₀.

Kategorie tematyczne

• 37F45: Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations
• 37F35: Conformal densities and Hausdorff dimension

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2011
• Tom: 214
• Numer: 2
• Strony: 119-133

Daty

wydano

2011

Twórcy

autor

Ludwik Jaksztas

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

Identyfikatory

DOI

10.4064/fm214-2-2