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Boundaries which arise in the dynamics of entire functions. (English) Zbl 0756.30021
R. 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.
Reviewer: I.N.Baker (London)

##### MSC:
 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable 37B99 Topological dynamics
##### Keywords:
boundary behaviour; Julia set; Cantor set; Jordan curve; prime end