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The Arnold cat map is employed in various applications where chaos is utilized, especially chaos-based cryptography and watermarking. In this paper, we study the problem of period distribution of the generalized discrete Arnold cat map over the Galois ring BBZ2e. Full knowledge of the period distribution is obtained analytically by adopting the Hensel lift approach. Our results have impact on both chaos theory and its applications as they not only provide design strategy in applications where special periods are required, but also help to identify unstable periodic orbits of the original chaotic cat map. The method in our paper also shows some ideas how to handle problems over the Galois ring BBZ2e.