The Julia set of the mapping \(z\to z\exp(z+\mu)\) (Q1334067)

In general a point \(z_ 0\) in the Julia set \(J(f)\) of a rational or entire function \(f\) is called a buried point of the Julia set if \(J(f)\) is not the whole complex plane and if \(z_ 0\) is not a boundary point of any component of the complement of \(J(f)\). Concerning the function \(f(z) = z \exp (z + \mu)\), where \(\mu\) is a parameter, the author proves: I. If \(J(f) \neq \mathbb{C}\) for real \(\mu \in [0,\infty)\) then \(J(f)\) is the closure of the set of buried points. In particular this occurs if \(\mu \in [0,2.5)\). II. There is an unbounded sequence of \(\mu\) values in \((2, \infty)\) such that \(J(f) = \mathbb{C}\).