Descending Periods in Compositions of Entire Functions

Source abstract

We consider a question of A. and C. Rényi concerning entire functions f and g for which f ◦ g is periodic. We isolate the case in which a period translation of f ◦ g descends through g to a conformal automorphism of g(C). In this descending regime we obtain a complete classification: either g is periodic, or g(z+T) = g(z) + K for some nonzero constant K, and f is K-periodic. Consequently, any example outside the periodic and translation cases must come from a non-descending period. The quadratic pullback examples lie in this remaining case.