Cycle Equivalence of Finite Dynamical Systems Containing Symmetries

Abstract

Two dynamical systems are cycle equivalent if they are topologically conjugate when restricted to their periodic points. In this paper, we extend our earlier results on cycle equivalence of asynchronous finite dynamical systems (FDSs) where the dependency graph may have a nontrivial automorphism group. We give conditions for when two update sequences \(\pi ,\pi '\) give cycle equivalent maps \(F_\pi , F_{\pi '}\), and we give improved upper bounds for the number of distinct cycle equivalence classes that can be generated by varying the update sequence. This paper contains a brief review of necessary background results and illustrating examples, and concludes with open questions and a conjecture.