Combinatorics of sequential dynamical systems

13 years 4 months ago

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In this paper we study sequential dynamical systems (SDS) over words. Our main result is the classification of SDS over words for fixed graphY and family of local maps (Fvi ) 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 graphY, (b) a family of vertex indexedY-local maps Fvi : Kn Kn, where K is a finite field and (c) a word w, i.e. a family (w1, . . . , wk), where wj is a Y-vertex. A map Fvi (xv1 , . . . , xvn ) is called Y-local iff it fixes all variables xvj = xvi and depends exclusively on the variables xvj , for vj B1(vi). The SDS-map is obtained by composing the local maps Fvi according to the word w: [(Fvi )viY , w] = k i=1Fwi : Kn - Kn. 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 ws, wr are adjacent inY. We pr...

Christian M. Reidys

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