Some examples of functions mapping rationals on rationals (Q1821900)

The author continues the study of analytic functions for which the set of all roots of unity is completely invariant [Sitzungsber., Abt. II, Österr. Acad. Wiss., Math.-Naturwiss. Kl. 195, 53-70 (1986; reviewed above)]. Typical results are the following: Theorem 1. Suppose that f is regular in \({\bar {\mathbb{C}}}\setminus E\) where \(E\subset \{z:| z| =1\}\) is a finite set. If for every root of unity w all solutions of \(f(z)=w\) are also roots of unity then \(f=const\). Theorem 2. For every \(K>1\) there exists f regular in \(\{0<| z| <\infty \}\), transcendental, and such that for all z in the annulus \(\{K^{-1}<| z| <K\}f(z)\) is a root of unity iff z is a root of unity.