Combinatorics of sequential dynamical systems

Abstract

In this paper we study sequential dynamical systems ( SDS) over words. Our main result is the classification of SDS over words for fixed graph Y and family of local maps ( F v i ) by means of a novel notion of SDS equivalence. This equivalence arises from a natural group action on acyclic orientations. An SDS consists of: (a) a graph Y, (b) a family of vertex indexed Y-local maps F v i : K n → K n , where K is a finite field and (c) a word w , i.e. a family ( w 1 , … , w k ) , where w j is a Y-vertex. A map F v i ( x v 1 , … , x v n ) is called Y-local iff it fixes all variables x v j ≠ x v i and depends exclusively on the variables x v j , for v j ∈ B 1 ( v i ) . The SDS-map is obtained by composing the local maps F v i according to the word w : [ ( F v i ) v i ∈ Y , w ] = ∏ i = 1 k F w i : K n ⟶ K n . Mutual dependencies of the local maps arising from their sequential application are expressed in the graph G ( w , Y ) having vertex set { 1 , … , k } (the indices of the word w ) and in which r , s are adjacent iff w s , w r are adjacent in Y. We prove a bijection from equivalence classes of SDS-words into equivalence classes of acyclic orientations of G ( w , Y ) . We show that within these equivalence classes the induced SDS are equivalent in the sense that their respective phase spaces are isomorphic as digraphs.