Geodesic
On the Periodicity of Compositions of Entire Functions
Canadian journal of mathematics, Tome 18 (1966) no. 1, pp. 724-730

Voir la notice de l'article provenant de la source Cambridge University Press

For two entire functions f(z) and g(z) the composition f(g(z)) may or may not be periodic even though g(z) is not periodic. For example, when f(u) = cos √u and g(z) = z2, or f(u) = eu and g(z) = p(z) + z, where p(z) is a periodic function of period 2πi, f(g(z)) will be periodic. On the other hand, for any polynomial Q(u) and any non-periodic entire function f(z) the composition Q(f(z)) is never periodic (2).The general problem of finding necessary and sufficient conditions for f(g(z)) to be periodic is a difficult one and we have not succeeded in solving it. However, we have found some interesting related results, which we present in this paper. 
Gross, Fred. On the Periodicity of Compositions of Entire Functions. Canadian journal of mathematics, Tome 18 (1966) no. 1, pp. 724-730. doi: 10.4153/CJM-1966-072-7
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