A self-invertibility condition for global periodicity of difference equations (Q2371112)

The author considers the global periodicity i.e. periodicity of all solutions with the same period \(p\) for the difference equation \[ y_{n+1}=f(y_n,\ldots,y_{n-k+1}),\quad k\geq 2 \] with the first order difference equation \[ z_{n+1}=g(z_n) \] which has global periodicity of prime period 1 or 2 if and only if \(g^{-1}(x)=g(x)\). In the general case an analogous condition appears to be only necessary for the global periodicity.