Additive conserved quantities in discrete-time lattice dynamical systems

Abstract

We give a necessary and sufficient condition for the existence of additive conserved quantities for one-dimensional discrete-time lattice dynamical systems such as cellular automata (CA) and coupled map lattices. Proof is given for the case of nearest-neighbor interaction rules. The obtained condition guarantees the existence of the current conservation law and is useful for finding additive conserved quantities for CA. Applications are made to Wolfram's elementary CA and elementary reversible CA of Takesue. In the latter case, the additive conserved quantities are classified into those induced by local conservation laws and propagative ones.