We initiate the study of effective pointwise ergodic theorems in resource-bounded settings. Classically, the convergence of the ergodic averages for integrable functions can be arbitrarily slow [Ulrich Krengel, 1978]. In contrast, we show that for a class of PSPACE L¹ functions, and a class of PSPACE computable measure-preserving ergodic transformations, the ergodic average exists and is equal to the space average on every EXP random. We establish a partial converse that PSPACE non-randomness can be characterized as non-convergence of ergodic averages. Further, we prove that there is a class of resource-bounded randoms, viz. SUBEXP-space randoms, on which the corresponding ergodic theorem has an exact converse - a point x is SUBEXP-space random if and only if the corresponding effective ergodic theorem holds for x.
@InProceedings{nandakumar_et_al:LIPIcs.MFCS.2021.80, author = {Nandakumar, Satyadev and Pulari, Subin}, title = {{Ergodic Theorems and Converses for PSPACE Functions}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {80:1--80:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.80}, URN = {urn:nbn:de:0030-drops-145204}, doi = {10.4230/LIPIcs.MFCS.2021.80}, annote = {Keywords: Ergodic Theorem, Resource-bounded randomness, Computable analysis, Complexity theory} }
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