Boundaries which arise in the dynamics of entire functions.

*(English)*Zbl 0756.30021R. L. Devaney and L. Goldberg [Duke Math. J. 55, 253-266 (1987; Zbl 0621.30024)] showed that entire functions \(f(z)=\lambda e^ z\), where \(\lambda=te^{-t}\), \(| t| <1\), have a Julia set \(J\), which is the union of a Cantor set of curves. Each curve in the set runs to \(\infty\). The set \(N(f)\) where iterates of \(f\) are normal is in this case a single domain whose boundary is \(J\).

In the present paper it is shown that if \(g\) is transcendental entire and \(G\) is an unbounded component \(N(g)\), then either (i) \(f^ n\to \infty\) in \(G\) or (ii) \(\infty\) belongs to the impression of every prime end of \(G\). Thus in case (ii) the impression of every prime end of \(G\) is (almost always) a continuum which includes \(\infty\); in the remaining cases it is just the point \(\infty\).

It is also shown that there are some \(g\) of the form \(g(z)=z+\gamma+\exp(2\pi iz)\), \(\gamma\) constant, for which there is an unbounded component \(G\) of \(N(g)\) such that \(\partial G\) is a Jordan curve.

In the present paper it is shown that if \(g\) is transcendental entire and \(G\) is an unbounded component \(N(g)\), then either (i) \(f^ n\to \infty\) in \(G\) or (ii) \(\infty\) belongs to the impression of every prime end of \(G\). Thus in case (ii) the impression of every prime end of \(G\) is (almost always) a continuum which includes \(\infty\); in the remaining cases it is just the point \(\infty\).

It is also shown that there are some \(g\) of the form \(g(z)=z+\gamma+\exp(2\pi iz)\), \(\gamma\) constant, for which there is an unbounded component \(G\) of \(N(g)\) such that \(\partial G\) is a Jordan curve.

Reviewer: I.N.Baker (London)