Area and Hausdorff dimension of the set of accessible points of the Julia sets of $λe^z$ and $λ \sin(z)$

Abstract

The Julia set Jof the exponential function E� : z ! �e z for � 2 (0,1/e) is known to be a union of curves (hairs) whose endpoints Care the only accessible points from the basin of attraction. We show that foras above the Hausdorff dimension of Cis equal to 2 and we give estimates for the Hausdorff dimension of the subset of Crelated to a finite number of symbols. We also consider the set of endpoints for the sine family F� : z ! (1/(2i))�(e iz e iz ) for � 2 (0,1) and prove that it has positive Lebesgue measure.