Periodic orbits in periodic discrete dynamics

We study the combinatorial structure of periodic orbits of nonautonomous difference equations x_{n + 1} = f_n (x_n) in a periodically fluctuating environment. We define the Γ-set to be the set of minimal periods that are not multiples of the phase period. We show that when the functions f_n are rational functions, the Γ-set is a finite set. In particular, we investigate several mathematical models of single-species without age structure, and find that periodic oscillations are influenced by periodic environments to the extent that almost all periods are divisors or multiples of the phase period.