# Certain Morphisms of Dynamical Systems

Wednesday, November 5, 2003 - 3:50pm - 4:25pm

Keller 3-180

Christian Reidys (Los Alamos National Laboratory)

We study a class of discrete dynamical systems that consists 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 We study a class of discrete dynamical systems that consists 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):F2n implyF2n)i and (c) a permutation p. The function F(i,Y) updates the binary state of vertex i as a function of the states of vertex i and its immediate Y-neighbors and leaves the states of all other vertices fixed. The permutation p 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 p we obtain the sequential dynamical system (SDS). Let G be the graph representing the phase space of the SDS. A SDS-morphism between the two SDS [FY,p] and [FZ,s] is a triple consisting of a graph morphism v: Y implyZ, a map e:Szimply Sy, where z,y denote the cardinalities of Z and Y, respectively such that e(s)=p and finally a digraph morphism between G(Z) and G(Y). Our main result is that any locally bijective graph morphisms (coverings) between dependency graphs of SDS naturally induce SDS-morphisms and we further give applications of this theorem that allow to identify variuos phases space properties of SDS.