Approximated periods of a mixing transformation

Abstract

A $Z_N$-linear transformation on $Z_N$ x $Z_N$ is defined by(x, y) $\rightarrow$ (x+y, x+2y) (mod N) for a given positive integer N. Let $m_N$ denote the period of the transformation. We showed that $m_{p^k}=m_p\cdot{p}^{k-1}$ where p is an odd prime and $m_p$ even.