Wyniki wyszukiwania

Sortuj według:

Ogranicz wyniki do:

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

100%

Jaksztas L.

Fundamenta Mathematicae

2011

tom 214

nr 2

119-133

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.

Infinite Iterated Function Systems Depending on a Parameter

100%

Jaksztas L.

Bulletin of the Polish Academy of Sciences. Mathematics

2007

tom 55

nr 2

105-122

This paper is motivated by the problem of dependence of the Hausdorff dimension of the Julia-Lavaurs sets $J_{0,σ}$ for the map f₀(z) = z²+1/4 on the parameter~σ. Using homographies, we imitate the construction of the iterated function system (IFS) whose limit set is a subset of $J_{0,σ}$, given by Urbański and Zinsmeister. The closure of the limit set of our IFS ${ϕ^{n,k}_{σ,α}}$ is the closure of some family of circles, and if the parameter σ varies, then the behavior of the limit set is similar to the behavior of $J_{0,σ}$. The parameter α determines the diameter of the largest circle, and therefore the diameters of other circles. We prove that for all parameters α except possibly for a set without accumulation points, for all appropriate t > 1 the sum of the tth powers of the diameters of the images of the largest circle under the maps of the IFS depends on the parameter σ. This is the first step to verifying the conjectured dependence of the pressure and Hausdorff dimension on σ for our model and for $J_{0,σ}$.

Ograniczanie wyników

1 Bulletin of the Polish Academy of Sciences. Mathematics

1 Fundamenta Mathematicae

2 Jaksztas L.

1 2011

1 2007

Wyniki wyszukiwania

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

Infinite Iterated Function Systems Depending on a Parameter