Area and Hausdorff dimension of Julia sets of entire functions

Abstract

We show the Julia set of λ sin ⁡ ( z ) \lambda \sin (z) has positive area and the action of λ sin ⁡ ( z ) \lambda \sin (z) on its Julia set is not ergodic; the Julia set of λ exp ⁡ ( z ) \lambda \exp (z) has Hausdorff dimension two but in the presence of an attracting periodic cycle its area is zero.