# Moscow Mathematical Journal

Volume 14, Issue 4, October–December 2014 pp. 711–744.

Randomness and Non-Ergodic Systems

**Authors**:
Johanna N.Y. Franklin (1) and Henry Towsner (2)

**Author institution:**(1) Department of Mathematics, Room 306, Roosevelt Hall, Hofstra University, Hempstead, NY 11549-0114, USA

(2) Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104-6395, USA

**Summary: **

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-Lӧf 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.

2010 Mathematics Subject Classification. Primary: 03D32; Secondary: 37A25

**Keywords:**Algorithmic randomness, Martin-Lӧf random, dynamical system, ergodic theorem, upcrossing

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