A new characterization of periodic oscillations in periodic difference equations

In this paper, we characterize periodic solutions of p-periodic difference equations. We classify the periods into multiples of p and nonmultiples of p. We show that the elements of the set of multiples of p follow the well-known Sharkovsky's ordering multiplied by p. On the other hand, we show that the elements of the set Γ_p of nonmultiples of p are independent in their existence. Moreover, we show the existence of a p-periodic difference equation with infinite Γ_p-set in which the maps are defined on a compact domain and agree exactly on a countable set. Based on the proposed classification, we give a refinement of Sharkovsky's theorem for periodic difference equations.