The Poincaré Recurrence Theorem

Abstract

In the course of his studies in celestial mechanics, Poincaré discovered a theorem which is remarkable both for its simplicity and for its far-reaching consequences. It is noteworthy also for having initiated the modern study of measure-preserving transformations, known as ergodic theory. From our point of view, this “recurrence theorem” has a special interest, because in proving it Poincaré anticipated the notions of both measure and category. Publication of his treatise, “Les méthodes nouvelles de la mécanique céleste” [29], antedated slightly the introduction of either notion.