Martin-Löf random points satisfy Birkhoff's ergodic theorem for effectively closed sets (Q2845878)

The main result of this paper (Theorem \(7\)) is a link between ergodic theory and algorithmic randomness, namely that for \((X,\mu)\) a probability space and \(T:X\rightarrow X\) a computable, measure-preserving transformation, every Poincaré point with respect to the \(\Pi_{1}^{0}\)-subsets of \(X\) is also a Birkhoff point for \(T\) with respect to the \(\Pi_{1}^{0}\)-subsets of \(X\). Here, \(x\in X\) is a Poincaré point of \(T\) with respect to a set \(Y\subseteq\mathfrak{P}(X)\) of measurable subsets of \(X\) when the sequence \(\{T^{i}(x) \mid i\in\omega\}\) of iterated applications of \(T\) to \(x\) has infinite intersection with each element of \(Y\) of positive measure; it is a Birkhoff point when it appears in each element of \(Y\) `appropriately often', i.e. when the asymptotic density of the set of \(n\) for which \(T^{n}(x)\in E\) is equal to the measure of \(E\) for each \(E\in Y\). Quite obviously, every Birkhoff point is a Poincaré point, but it is a pleasant surprise that in the special circumstances of this paper, the converse holds as well. A nice consequence of this is Theorem \(6\), a new characterization of the Martin-Löf random elements \(x\) of \(X\) as the Birkhoff points for \(T\) with respect to the \(\Pi_{1}^{0}\)-subsets of \(X\) whenever \(T\) is computable and ergodic. This neatly complements the earlier characterization of Schnorr random points as Birkhoff points for computable ergodic transformations with respect to \(\Pi_{1}^{0}\)-sets of computable measure by Gacs, Hoyrup and Rojas.NEWLINENEWLINE The paper is carefully written and, especially with respect to its length of \(6\) pages, pleasantly self-contained. Accordingly, it should be accessible to anyone who understands the result.