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 Mosc. Math. J., 2014, Volume 14, Number 4, Pages 711–744 (Mi mmj542)

Randomness and non-ergodic systems

Johanna N. Y.  Franklina, Henry Towsnerb

a Department of Mathematics, Room 306, Roosevelt Hall, Hofstra University, Hempstead, NY 11549-0114, USA
b Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104-6395, USA

Abstract: We characterize the points that satisfy Birkhoff's ergodic theorem under certain computability conditions in terms of algorithmic randomness. First, we use the method of cutting and stacking to show that if an element $x$ of the Cantor space is not Martin-Löf random, there is a computable measure-preserving transformation and a computable set that witness that $x$ is not typical with respect to the ergodic theorem, which gives us the converse of a theorem by V'yugin. We further show that if $x$ is weakly $2$-random, then it satisfies the ergodic theorem for all computable measure-preserving transformations and all lower semi-computable functions.

Key words and phrases: algorithmic randomness, Martin-Löf random, dynamical system, ergodic theorem, upcrossing.

DOI: https://doi.org/10.17323/1609-4514-2014-14-4-711-744

Full text: http://www.mathjournals.org/.../2014-014-004-004.html
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MSC: Primary 03D32; Secondary 37A25
Received: June 18, 2012; in revised form January 22, 2014
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Citation: Johanna N. Y.  Franklin, Henry Towsner, “Randomness and non-ergodic systems”, Mosc. Math. J., 14:4 (2014), 711–744

Citation in format AMSBIB
\Bibitem{FraTow14} \by Johanna~N.~Y.~~Franklin, Henry~Towsner \paper Randomness and non-ergodic systems \jour Mosc. Math.~J. \yr 2014 \vol 14 \issue 4 \pages 711--744 \mathnet{http://mi.mathnet.ru/mmj542} \crossref{https://doi.org/10.17323/1609-4514-2014-14-4-711-744} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3292047} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000349324800004} 

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• http://mi.mathnet.ru/eng/mmj/v14/i4/p711

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. K. Miyabe, A. Nies, J. Zhang, “Using almost-everywhere theorems from analysis to study randomness”, Bull. Symb. Log., 22:3 (2016), 305–331