The recurrence theorem is valid for volume-preserving flows on Riemannian manifolds $ V $ of finite volume. The recurrence theorem is also true for a discrete-time dynamical system, e.g. for a mapping $ f $ of a bounded domain in Euclidean space to itself that preserves Lebesgue measure. See for another generalization. On a content level Poincare theorem states for any open set GC X points, returning relatively G, are all points G, except for some set of the first category on measure zero. Formally takes place Theorem 1 [5]. Let - be a measure-preserving transformation of a probability space (X ,P) and let A X be a measurable set. sq g btoprezr — $0 abgcg Oil pi V; El-roe s $0 pa 130 ü3eee co oc {:yG gyg {:psas ap014: pa batroqa ayorrrq pa acobTc; we n;gcr-c— bysae Poincaré Recurrence Theorem. Related to the concept of eternal return is the Poincaré recurrence theorem in mathematics. It states that a system whose dynamics are volume-preserving and which is confined to a finite spatial volume will, after a sufficiently long time, return to an arbitrarily small neighborhood of its initial state. Looking for Poincarè recurrence theorem? Find out information about Poincarè recurrence theorem. A volume preserving homeomorphism T of a finite dimensional Euclidean space will have, for almost all points x , infinitely many points of the form T i , i Poincaré recurrence theorem. In mathematics, the Poincaré recurrence theorem states that certain systems will, after a sufficiently long but finite time, return to a state very close to the initial state. The Poincaré recurrence time is the length of time elapsed until the recurrence (this time may vary greatly depending on the exact initial Physics and philosophy are two subjects that have always been closely linked. The Eternal Return is one of the most extraordinary concepts in the philosophy However, the application of the Poincaré recurrence theorem Poincare , see below, gave rise to Zermelo's Zermelo paradox, which has not been resolved to everyone's satisfaction yet, and is the subject matter of this work. The recurrence theorem is valid for a classical system and basically states that provided an isolated mechanical system, in which the forces do not depend on the number theory. For example we will see that van der Waerden's theorem on arithmetic progressions is a consequence of an appropriate generalization of Birkhoff's recurrence theorem. A more recent result is that of Szemerédi stating that a subset of the integers having positive upper density contains arbitrarily long arithmetic progressions.