On the dynamics of entire functions on the Julia set

Abstract

Let f be an entire function and ƒn its n-th iterate. Let P(ƒ) denote the postcritical set of ƒ and J(ƒ) the Julia set of ƒ. Suppose that the set E of all z ∈ J(ƒ) with limsupn→∞ dist (ƒn(z), P(ƒ) U {∞}) > 0 has positive measure. It is proved that for a given set A ⊆ ℂ of positive measure the set {n ∈ ℕ; ƒn(z) ∈ A} is infinite for almost all z in the plane. From this follows that the forward orbit of almost all z ∈ ℂ is dense in the plane if E has positive measure.