Exponential polynomials with Fatou and non-escaping sets of finite Lebesgue measure

Abstract

We give conditions ensuring that the Fatou set and the complement of the fast escaping set of an exponential polynomial f both have finite Lebesgue measure. Essentially, these conditions are designed such that $|f(z)|\ge \exp (|z|^\alpha )$ for some $\alpha>0$ and all z outside a set of finite Lebesgue measure.

Transcendental Julia sets of positive finite Lebesgue measure

“…Schubert [10] showed that the Lebesgue measure of the Fatou set of sin(z) is finite in any vertical strip of width 2π. Hemke [7] and the author [13] gave examples of transcendental entire functions whose Fatou set has finite measure. Now the natural question arises whether the Julia set of a transcendental entire function can also have positive measure and still be small in the sense that the Julia set itself has finite measure.…”