The Poincaré recurrence theorem states that probabilistic measure-preserving dynamical systems will, after a sufficiently long but finite time, return to a state arbitrarily close to their initial state. This raises a natural question: How long must one wait? The rate of recurrence can be quantified in cases where the space has a finite Hausdorff dimension. We will prove the classical Poincaré recurrence theorem and its quantitative version due to Boshernitzan. Some applications will be provided. This talk is will be very accessible; no prior knowledge is required.
Reference:
[B] Boshernitzan, M.D. Quantitative recurrence results. Invent. Math. 113, 617–631 (1993).