Martin-Löf random points satisfy Birkhoff’s ergodic theorem for effectively closed sets

Abstract

Abstract. We show that if a point in a computable probability space X satisfies the ergodic recurrence property for a computable measure-preserving T : X → X with respect to effectively closed sets, then it also satisfies Birkhoff's ergodic theorem for T with respect to effectively closed sets. As a corollary, every Martin-Löf random sequence in the Cantor space satisfies Birkhoff's ergodic theorem for the shift operator with respect to Π 0 1 classes. This answers a question of Hoyrup and Rojas.Several theorems in ergodic theory state that almost all points in a probability space behave in a regular fashion with respect to an ergodic transformation of the space. For example, if T : X → X is ergodic, 1 then almost all points in X recur in a set of positive measure:. Let (X, μ) be a probability space, and let T : X → X be ergodic. For all E ⊆ X of positive measure, for almost all x ∈ X, T n (x) ∈ E for infinitely many n.

Computability and analysis: the legacy of Alan Turing

“…Indeed, the work of Gács, Galatolo, Hoyrup, and Rojas [68,67], together with the results of [8], shows that the Schnorr random reals are exactly the ones at which the ergodic theorem holds with respect to every ergodic measure and computable measure-preserving transformation (see also the discussion in [166]). Bienvenu, Day, Hoyrup, and Shen [15] and, independently, Franklin, Greenberg, Miller, and Ng [59] have extended V'yugin's result to the characteristic function of a computably open set, in the case where the measure is ergodic.…”

On zeros of Martin-Löf random Brownian motion

“…We shall prove the following. We can now apply the effective ergodic theorem proven in Bienvenu et al [3] and independently Franklin et al [10]: since U has measure less than 1 (by Kahane's theorem) and is a Σ 0 1 set, there are infinitely many n such that S n (B) / ∈ U (in fact, the set of such n's is a subset of N of positive density), ie such that…”

Randomness, Computation and Mathematics

“…al. [33] looked at the classic Birkhoff ergodic theorem for f ∈ L 1 (X) (namely lim n→∞ 1 n i<n f (T i (x)) = f dµ.) and showed that 1-random points satisfy Birkhoff's ergodic theorem.…”