Deficiency for meromorphic solutions of schröder equations

Abstract

Eremenko and Sodin proved that meromorphic solution f (z) of the Schröder equation f (sz) = R (f (z)), |s| > 1, has no Valiron deficiency other than exceptional values of R(z). We consider transcendental meromorphic solutions of non-autonomous equation f (sz) =R (z, f (z)), |s| > 1. It is shown that there exists an equation of this form possessing a transcendental meromorphic solution, which has a Valiron deficiency other than a Nevanlinna deficiency. We also give some generalizations of the Eremenko and Sodin theorem for algebraic functions as targets.