Sequential Dynamical Systems Over Words

Abstract.In this paper we study sequential dynamical systems (SDS) over words. An SDS is a triple consisting of: (a) a graph Y with vertex set {v1, ..., vn}, (b) a family of Y-local functions $$ (F_{v_{i}})_{1 \leq i \leq n} , F_{v_{i}} :K^{n} \to K^{n} $$, where K is a finite field and (c) a word w, i.e., a family (w1, ..., wk), where wi is a Y-vertex. A map $$ F_{v_{i}} (x_{v_{1}} , \ldots x_{v_{n}} ) $$ is called Y-local if and only if it fixes all variables $$ x_{v_{j}} \ne x_{v_{i}} $$ and depends exclusively on the variables $$ x_{v_{j}} $$ , for $$ v_{j} \in B_{1} (v_{i} ) $$. An SDS induces an SDS- map, $$ {\left[ {(F_{v_{i}} )_{{v_{i} \in Y}} ,w} \right]} = {\prod\nolimits_{i = 1}^k {F_{w_{i}} :K^{n} } } \to K^{n} $$, obtained by the map-composition of the functions $$ F_{v_{i}} $$ according to w. We show that an SDS induces in addition the graph G(w,Y) having vertex set {1, ..., k} where r, s are adjacent if and only if ws, wr are adjacent in Y. G(w, Y) is acted upon by Sk via $$ \rho \cdot w = (w_{{\rho ^{{ - 1}} (1)}} , \ldots w_{{\rho ^{{ - 1}} (k)}} ) $$ and Fix(w) is the group of G(w, Y) graph automorphisms which fix w. We analyze G(w, Y)-automorphisms via an exact sequence involving the normalizer of Fix(w) in Aut(G(w, Y)), Fix(w) and Aut(Y). Furthermore we introduce an equivalence relation over words and prove a bijection between word equivalence classes and certain orbits of acyclic orientations of G(w, Y).