The dynamics of a linear dynamical system over a finite field can be described by using the elementary divisors of the corresponding matrix. It is natural to extend the investigation to a general finite commutative ring. In a previous publication, the last two authors developed an efficient algorithm to determine whether a linear dynamical system over a finite commutative ring is a fixed point system or not. The algorithm can also be used to reduce the problem of finding the cycles of such a system to the case where the system is given by an automorphism. Here, we further analyze the cycle structure of such a system and develop a method to determine its cycles.
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The assumption on the module considered in Theorem 2.1  is somewhat different, but a quick check of the proof there reveals that it also applies to the case here.
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Part of the work in this paper was done during the visit of Y.J.W. to the Department of Mathematical Sciences at the University of Wisconsin-Milwaukee in the summer of 2015. Y.J.W. wishes to thank the University of Wisconsin-Milwaukee and its faculty for the hospitality she received during her visit.
Y.J.W. acknowledges the support of the National Natural Science Foundation of China (11461010) and the Guangxi Natural Science Foundation (2014GXNSFAA118005).
G.W.X. acknowledges partial support from the National 973 Project of China (2013CB834205).
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Wei, Y., Xu, G. & Zou, Y.M. Dynamics of linear systems over finite commutative rings. AAECC 27, 469–479 (2016). https://doi.org/10.1007/s00200-016-0290-y