Randomness and non-ergodic systems

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.