Chaos, entropy and integrals for discrete dynamical systems on lattices

Abstract

Discrete time dynamical systems on discrete state spaces called lattices are the subject of this paper. These dynamical systems have properties that differ from discrete time dynamical systems on continuous state spaces. These differences include the existence of discrete integrals as constants of motion along orbits. Computer arithmetic, on lattices of floating point numbers, is used typically to evaluate orbits of discrete dynamical systems defined on continuous state spaces, i.e. the iteration of maps on continuous spaces. For chaotic dynamical systems, in which orbits diverge rapidly, computer arithmetic is not suited to this purpose. Calculations based on exact arithmetic performed on lattices are developed in this paper. New dynamical properties are found for discrete dynamical systems on discrete state spaces by restricting to embedded lattices discrete dynamical systems on continuous state spaces. The new results include theorems on existence of discrete integrals for discrete time dynamical systems on lattices, entropy of orbits described by probability distributions, and decompositions of lattices by the dynamics into orbits. A new viewpoint emerges when one takes a countable infinite union of lattices. This is a way in which the computable number field can be approached. The nature of dynamics changes when the discrete state space is infinite. Chaos appears naturally in this setting and in an interesting way.