This article is cited in 1 scientific paper (total in 1 paper)
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
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.
MSC: Primary 03D32; Secondary 37A25
Received: June 18, 2012; in revised form January 22, 2014
Johanna N. Y. Franklin, Henry Towsner, “Randomness and non-ergodic systems”, Mosc. Math. J., 14:4 (2014), 711–744
Citation in format AMSBIB
\by Johanna~N.~Y.~~Franklin, Henry~Towsner
\paper Randomness and non-ergodic systems
\jour Mosc. Math.~J.
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K. Miyabe, A. Nies, J. Zhang, “Using almost-everywhere theorems from analysis to study randomness”, Bull. Symb. Log., 22:3 (2016), 305–331
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