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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 Π0n-genericity and quasi-Π0n-genericity. We also provide conditions under which points recur at a positive frequency in sets containing them, regarding both simple and multiple recurrence.
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BibTeX
@misc{veltri2026effective,
title = {{Effective recurrence for computable measure-preserving transformations}},
author = {Joey Veltri},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12402},
eprint = {2609.12402},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12402},
}
RIS
TY - UNPB
TI - Effective recurrence for computable measure-preserving transformations
AU - Joey Veltri
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12402
UR - https://arxiv.org/abs/2609.12402
AB - 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 Π^0_n-genericity and quasi-Π^0_n-genericity. We also provide conditions under which points recur at a positive frequency in sets containing them, regarding both simple and multiple recurrence.
ER -