On certain morphisms of sequential dynamical systems

Abstract

We study a class of discrete dynamical systems that consist of the following data: (a) a finite (labeled) graph Y with vertex set { 1 , … , n } , where each vertex has a binary state, (b) a vertex labeled multi-set of functions ( F i , Y : F 2 n → F 2 n ) i and (c) a permutation π ∈ S n . The function F i , Y updates the binary state of vertex i as a function of the states of vertex i and its Y-neighbors and leaves the states of all other vertices fixed. The permutation π represents a Y-vertex ordering according to which the functions F i , Y are applied. By composing the functions F i , Y in the order given by π we obtain the sequential dynamical system (SDS) [ F Y , π ] = ∏ i = 1 n F π ( i ) , Y : F 2 n ⟶ F 2 n . Let G [ F Y , π ] be the graph with vertex set F 2 n and edge set { ( x , [ F Y , π ] ( x ) ) ∣ x ∈ F 2 n } . An SDS-morphism between [ F Y , π ] and [ F Z , σ ] is a triple ( ϕ , η , Φ ) , where ϕ : Y ⟶ Z is a graph-morphism, η : S | Z | ⟶ S | Y | is a map such that η ( σ ) = π and Φ is a digraph-morphism Φ : G [ F Z , σ ] ⟶ G [ F Y , π ] . Our main result is that locally bijective graph-morphisms (coverings) between dependency graphs of SDS naturally induce SDS-morphisms. We show how these SDS-morphisms allow for a new proof for the upper bound on the number of inequivalent SDS obtained by only varying their underlying permutations. Here, two SDS are called inequivalent if they are inequivalent as dynamical systems. Furthermore, we apply our result in order to obtain phase space properties of SDS.