Computing the Cycle Structure of Finite Linear Systems

Abstract Consider a linear difference equation with constant coefficients in the ring of integers modulo m. If the leading coefficient and the constant term are both units, then all trajectories are (purely) periodic. Moreover, the finite state set can be decomposed into disjoint cycles of various lengths. The following problems will be addressed: computing the cycle partition and determining the period w.r.t. a specific initial state. The latter question can often be reduced to calculating the order of an invertible matrix. If the prime factorization of m is known, then it suffices to consider prime powers, by the Chinese remainder theorem. For primes, an efficient algorithm due to Leedham-Green may be used, which is available in group-theoretic computer algebra systems such as Magma or GAP. This approach will be extended to prime powers. Finally, we will discuss how to relax the assumptions guaranteeing periodicity.