Cycle Equivalence of Sequential Dynamical Systems

A graph dynamical system consists of an undirected graph Y where each vertex is assigned a state from a finite set K, an update function for each vertex that depends only on the state of the vertex and its neighbors, and a method that dictates how to compose these functions. A common subclass of graph dynamical systems are sequential dynamical systems, or SDSs, where the update functions are applied sequentially, according to a given update order. An SDS where each vertex is updated precisely once at each step is called a permutation SDS. Two graph dynamical systems are cycle equivalent if their phase spaces are isomorphic when restricted to their sets of periodic points. In this paper we characterize cycle equivalence of SDSs as a function of shifts and reflections of the update order. Shifts of update orders for SDSs are shown to be closely related to conjugation of Coxeter elements, and we derive new graph constructions to obtain combinatorial bounds for the number of cycle equivalence classes of SDSs. The case of K = F2 arises frequently in the study of discrete, finite dynamical systems, and we show that when K = F2, update order reflections give rise to additional new combinatorial structures with no direct analog in Coxeter theory. As particular cases we derive upper bounds on the number of cycle equivalence classes for trees, circle graphs and graphs that are vertex joins such as the wheel graphs. As a corollary about dynamics, we prove that if the graph Y is a union of trees, then all permutation SDSs over Y are cycle equivalent for any fixed choice of vertex functions. This work extends existing work on functional equivalence of SDSs, and also extends aspects of existing work on enumeration of conjugacy classes of Coxeter elements.