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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Martin-Löf random points satisfy Birkhoff's ergodic theorem for effectively closed sets


Authors: Johanna N. Y. Franklin, Noam Greenberg, Joseph S. Miller and Keng Meng Ng
Journal: Proc. Amer. Math. Soc. 140 (2012), 3623-3628
MSC (2010): Primary 03D22; Secondary 28D05, 37A30
Published electronically: February 20, 2012
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Abstract: We show that if a point in a computable probability space $ X$ satisfies the ergodic recurrence property for a computable measure-preserving $ T\colon X\to X$ with respect to effectively closed sets, then it also satisfies Birkhoff's ergodic theorem for $ T$ with respect to effectively closed sets. As a corollary, every Martin-Löf random sequence in the Cantor space satisfies Birkhoff's ergodic theorem for the shift operator with respect to $ \Pi ^0_1$ classes. This answers a question of Hoyrup and Rojas.


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Additional Information

Johanna N. Y. Franklin
Affiliation: Department of Mathematics, 6188 Kemeny Hall, Dartmouth College, Hanover, New Hampshire 03755-3551
Email: johannaf@gauss.dartmouth.edu

Noam Greenberg
Affiliation: School of Mathematics, Statistics, and Operations Research, Victoria University of Wellington, P.O. Box 600, Wellington, 6140 New Zealand
Email: greenberg@mcs.vuw.ac.nz

Joseph S. Miller
Affiliation: Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin 53706-1388
Email: jmiller@math.wisc.edu

Keng Meng Ng
Affiliation: Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin 53706-1388
Email: selwynng@math.wisc.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11179-7
PII: S 0002-9939(2012)11179-7
Received by editor(s): July 20, 2010
Received by editor(s) in revised form: April 5, 2011, and April 8, 2011
Published electronically: February 20, 2012
Additional Notes: The second author was partially supported by the Marsden Grant of New Zealand
The third author was supported by the National Science Foundation under grants DMS-0945187 and DMS-0946325, the latter being part of a Focused Research Group in Algorithmic Randomness.
Communicated by: Julia Knight
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.