Abstract
AbstractWe 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.
Highlights
Introduction and resultsLet f be a transcendental entire function, and let f n denote the nth iterate of f
The Fatou set F ( f ) of f is the set of all z ∈ C such that the iterates ( f n)n∈N form a normal family in a neighbourhood of z, and the Julia set J (f ) is the complement of F (f )
These sets have an important role in complex dynamics
Summary
The Lebesgue measure of C\(A(f ) ∩ J (f )) is finite. The functions (2) considered by Hemke satisfy the assumptions of Theorem 1.4. Throughout the rest of the paper, let f be an entire function satisfying the assumptions of Theorem 1.4. We finish the proof of Theorem 1.4 in §4, using a construction similar to one that was used in McMullen’s paper [9] and has since been used by various authors. 2. The behaviour of f we prove several properties of the function f.
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