Determining steady state behaviour of discrete monomial dynamical systems

In previous work [3] Colón-Reyes et al developed criteria for determining when a discrete monomial dynamical system reaches steady state behaviour. These criteria depend on determining when a certain matrix over a finite ring, that is not a field, defines a fixed point system. It was not until recently that criteria to determine linear steady state behaviour over rings have been found. Using these new results we present a new algorithm to determine steady state behaviour of monomial dynamical systems over finite fields. Delgado-Eckert [5] has also obtained an algorithm for the finite field case, but his algorithm does not take into account the result in [3] and requires $O(n^4\; q^2 \log\; q)$ integer operations. Our algorithm requires only $O(n^3 \log(n\; \log \; q))$ integer operations.