Poincare recurrence and number theory pdf

Poincares recurrence theorem is the first and most basic theorem of ergodic. The comparison between the scaling of the recurrence time with n performed above show that poincare recurrences in chaotic systems occur only after a very long time. Pdf the many facets of poincare recurrence theorem of. Concerning the second version of the poincare recurrence theorem, it is.

Although somewhat differently formulated, essentially the same objection was made in 1896 by plancks student e zermelo, who noted that the htheorem is inconsistent with poincare s recurrence theorem proved in 1890 and stated that any physical system, even with irreversible thermodynamic processes. Nonrecurrence sets for weakly mixing linear dynamical. Poincare recurrence and number theory nu math sites. This is in agreement with the usual assumptions of statistical physics when describing thermodynamical systems. In physics, the poincare recurrence theorem states that certain systems will, after a sufficiently long but finite time, return to a state arbitrarily close to for continuous state systems, or exactly the same as for discrete state systems, their initial state the poincare recurrence time is the length of time elapsed until the recurrence. Volume 5, number 3, november 1981 poincare recurrence and number theory by harry furstenberg introduction. Poincare recurrence and number theory project euclid. The fundamental theorem of algebra and complexity theory by steve smale 155 section 7. We begin by stating two famous numbertheoretic results that have been. Ergodic theory and recurrence poincare recurrence and number theory by harry furstenberg 193 the ergodic theoretical proof of szemeredis theorem by h. The proof of khintchines recurrence theorem uses the hubert space theory of l2x. The purpose of this paper is to illustrate the many aspects of poincare recurrence time theorem for an archetype of a complex system, the logistic map.

Hillel furstenbergs 1981 article in the bulletin gives an elegant. Now we all know this theorem is very useful in many areas of physics, especially statistical mechanics, but here arnold is really stressing that it also has an abstract value. The main focus will be on smooth dynamical systems, in particular, those with some chaotichyperbolic beha. Poincare recurrence and number theory thirty years later bryna kra hillel furstenbergs 1981 article in the bulletin gives an elegant introduction to the interplay between dynamics and number theory, summarizing the major developments that occurred in the few years after his landmark paper 21. Witnessing a poincare recurrence with mathematica sciencedirect. Ergodic theory studies properties of mpts that have to do with iteration of t. We present some recurrence results in the context of ergodic theory and dynamical systems.