ON GLOBAL PERIODICITY OF DIFFERENCE EQUATIONS

Abstract

We study the global periodicity of difference equations of the form $% x_{n+1}=f(x_n)$, $x_{n+1}=f_n(x_n)$ and $x_{n+2}=f(x_n,x_{n+1})$. We characterize the $n$-cycles %%%(all the solutions of the equation is periodic with period at most $n$) in the case of first order equations and give some partial results for the second order equation. In particular, we find some examples of $3$-cycles which are different from the equation $x_{n+2}=\frac c{x_n x_{n+1}}$, solving a question of [2] and [4].