Periodic properties of multidimensional polynomial generator over the Galois ring. I

Periodic properties of multidimensional polynomial substitutions over the Galois ring are investigated. Maximum $L_m(R)$ of cycle lengths of $m$-dimensional polynomial substitutions is computed. A method permitting to construct substitutions with cycle of length $L_m(R)$ is suggested. For a particular case the cycle type of $m$-dimensional polynomial substitution is founded. The paper generalizes earlier results to the case of arbitrary dimension $m$ and arbitrary Galois ring $R$.