On Petrenko's deviations and the Julia limiting directions of solutions of complex differential equations

Abstract

The Julia set J(f) of a transcendental meromorphic function f has chaotic behavior in the complex dynamics. The ray arg⁡z=θ∈[0,2π) is said to be a limiting direction of J(f) if there is an unbounded sequence {zn}⊆J(f) such that limn→∞⁡arg⁡zn=θ. In this paper, we mainly investigated the Julia limiting directions of entire solutions of complex differential equations, of which coefficients are associated with Petrenko's deviations. In fact, we proved the lower bounds of the sets of Julia limiting directions of these solutions have closed relations with Petrenko's deviations of the coefficients.