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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