Randomness and Non-ergodic Systems

TL;DR

This paper explores the relationship between algorithmic randomness and ergodic theory, showing how different levels of randomness influence typicality under computable measure-preserving transformations.

Contribution

It establishes a characterization of points satisfying Birkhoff's ergodic theorem based on computability and randomness levels, extending previous theorems.

Findings

Non-Martin-Löf random points are non-typical for some computable transformation.

Weakly 2-random points satisfy the ergodic theorem universally for computable transformations.

Provides a converse to V'yugin's theorem using cutting and stacking methods.

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-Lof 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.