Area and Hausdorff dimension of the set of accessible points of the Julia sets of $λe^z$ and $λ \sin(z)$
Volume 159 / 1999
Fundamenta Mathematicae 159 (1999), 269-287
DOI: 10.4064/fm_1999_159_3_1_269_287
Abstract
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.
