Area and Hausdorff dimension of the set of accessible points of the Julia sets of $λe^z$ and $λ \sin(z)$
Fundamenta Mathematicae, Tome 159 (1999) no. 3, pp. 269-287
The Julia set $J_λ$ of the exponential function $ E_λ:z → λ e^z$ for $λ ∈ (0,1/e)$ is known to be a union of curves ("hairs") whose endpoints $C_λ$ are the only accessible points from the basin of attraction. We show that for λ as above the Hausdorff dimension of $C_λ$ is equal to 2 and we give estimates for the Hausdorff dimension of the subset of $C_λ$ related to a finite number of symbols. We also consider the set of endpoints for the sine family $F_λ:z → (1/(2i))λ (e^{iz}-e^{-iz})$ for $λ ∈ (0,1)$ and prove that it has positive Lebesgue measure.
@article{10_4064_fm_1999_159_3_1_269_287,
author = {Bogus{\l}awa Karpi\'nska},
title = {Area and {Hausdorff} dimension of the set of accessible points of the {Julia} sets of $\ensuremath{\lambda}e^z$ and $\ensuremath{\lambda} \sin(z)$},
journal = {Fundamenta Mathematicae},
pages = {269--287},
year = {1999},
volume = {159},
number = {3},
doi = {10.4064/fm_1999_159_3_1_269_287},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/fm_1999_159_3_1_269_287/}
}
TY - JOUR AU - Bogusława Karpińska TI - Area and Hausdorff dimension of the set of accessible points of the Julia sets of $λe^z$ and $λ \sin(z)$ JO - Fundamenta Mathematicae PY - 1999 SP - 269 EP - 287 VL - 159 IS - 3 UR - http://geodesic.mathdoc.fr/articles/10.4064/fm_1999_159_3_1_269_287/ DO - 10.4064/fm_1999_159_3_1_269_287 LA - en ID - 10_4064_fm_1999_159_3_1_269_287 ER -
%0 Journal Article %A Bogusława Karpińska %T Area and Hausdorff dimension of the set of accessible points of the Julia sets of $λe^z$ and $λ \sin(z)$ %J Fundamenta Mathematicae %D 1999 %P 269-287 %V 159 %N 3 %U http://geodesic.mathdoc.fr/articles/10.4064/fm_1999_159_3_1_269_287/ %R 10.4064/fm_1999_159_3_1_269_287 %G en %F 10_4064_fm_1999_159_3_1_269_287
Bogusława Karpińska. Area and Hausdorff dimension of the set of accessible points of the Julia sets of $λe^z$ and $λ \sin(z)$. Fundamenta Mathematicae, Tome 159 (1999) no. 3, pp. 269-287. doi: 10.4064/fm_1999_159_3_1_269_287
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