Analys- och sannolikhetsseminarium

Abstrakt finns enbart på engelska: A fundamental result in dynamical systems is the Poincaré Recurrence Theorem, which states that, for a measure-preserving dynamical system, almost every point is recurrent. That is, almost every point returns infinitely often to a shrinking neighbourhood of its initial position. Although the theorem is striking in that it offers a strong qualitative conclusion under minimal assumptions, it leaves open the question of quantitative recurrence. For example, can one say something about how quickly the neighbourhoods may shrink, or about the rate at which the point returns to its initial position?

In this talk, I will present dynamical strong Borel--Cantelli lemmas (a sort of non-autonomous strong law of large numbers) that give precise asymptotic formulas for the rate of recurrence in terms of the measures of the shrinking neighbourhoods. The proofs boil down to a second moment analysis in which we use decay of correlations estimates -- a standard tool in establishing statistical laws for dynamical systems.

This talk is based on joint work with Tomas Persson.