Effective recurrence for computable measure-preserving transformations

Abstract

We prove several necessary and sufficient conditions under which a point satisfies the Poincaré Recurrence Theorem for all computable (ergodic) measure-preserving transformations and all sets of a particular complexity. The necessary conditions are obtained by constructing specific measure-preserving transformations which violate recurrence. While some of these conditions pertain to standard notions of algorithmic randomness, others involve new notions of genericity developed in the context of a computable probability space, which we call ${\Pi }_n^0$-genericity and quasi-${\Pi }_n^0$-genericity. We also provide conditions under which points recur at a positive frequency in sets containing them, regarding both simple and multiple recurrence.

65 pages, 4 figures