Period Distribution of Generalized Discrete Arnold Cat Map for $N=p^{e}$

In this paper, we analyze the period distribution of the generalized discrete cat map over the Galois ring where is a prime. The sequences generated by this map are modeled as 2-dimensional LFSR sequences. Employing the generation function and the Hensel lifting approaches, full knowledge of the detail period distribution is obtained analytically. Our results not only characterize the period distribution of the cat map, which gives insights to various applications, but also demonstrate some approaches to deal with the period of a polynomial in the Galois ring.