Poincare recurrence and number theory pdf

This is in agreement with the usual assumptions of statistical physics when describing thermodynamical systems. Poincare is largely responsible for the transformation of celestial mechanics from the study of individual solutions of differential equations to the global analysis of phase space. An application of poincares recurrence theorem to academic administration created date. In his recurrence theorem, poincare demonstrated how measuretheoretic ideas, particularly the idea of a measurepreserving group of transformations, lead to the. Recurrence in ergodic theory and combinatorial number theory. 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. Ergodic theory studies properties of mpts that have to do with iteration of t. 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. 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. Poincare recurrence and number theory nu math sites. We begin by stating two famous numbertheoretic results that have been. 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. 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. Concerning the second version of the poincare recurrence theorem, it is.

The proof of khintchines recurrence theorem uses the hubert space theory of l2x. The fundamental theorem of algebra and complexity theory by steve smale 155 section 7. We present some recurrence results in the context of ergodic theory and dynamical systems. 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. Volume 5, number 3, november 1981 poincare recurrence and number theory by harry furstenberg introduction. Witnessing a poincare recurrence with mathematica sciencedirect.