PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000–000 S 0002-9939(XX)0000-0
MARTIN-LOF¨ RANDOM POINTS SATISFY BIRKHOFF’S ERGODIC THEOREM FOR EFFECTIVELY CLOSED SETS
JOHANNA N.Y. FRANKLIN, NOAM GREENBERG, JOSEPH S. MILLER, AND KENG MENG NG
Abstract. We show that if a point in a computable probability space X sat- isfies 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-L¨ofrandom sequence in the Cantor space satisfies 0 Birkhoff’s ergodic theorem for the shift operator with respect to Π1 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: Theorem 1 (See [5]). 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. Recent investigations in the area of algorithmic randomness relate the hierarchy of notions of randomness to the satisfaction of computable instances of ergodic theorems. This has been inspired by Kuˇcera’sclassic result characterising Martin- L¨ofrandomness in the Cantor space. We reformulate Kuˇcera’sresult using the general terminology of [4]. Definition 2. Let (X, µ) be a probability space, and let T : X → X be a function. Let C be a collection of measurable subsets of X. A point x ∈ X is a Poincar´e point for T with respect to C if for all E ∈ C of positive measure for infinitely many n, T n(x) ∈ E. The Cantor space 2ω is equipped with the fair-coin product measure λ. The shift operator σ on the Cantor space is the function σ(a0a1a2 ... ) = a1a2 ... . The shift operator is ergodic on (2ω, λ).
Received by the editors April 10, 2011. The second author was partially supported by the Marsden grant of New Zealand. The third author was supported by the National Science Foundation under grants DMS-0945187 and DMS- 0946325, the latter being part of a Focused Research Group in Algorithmic Randomness. 1Recall that if (X, µ) is a probability space, then a measurable map T : X → X is measure preserving if for all measurable A ⊆ X, µ `T −1A´ = µ(A). We say that a measurable set A ⊆ X is invariant under a map T : X → X if T −1A = A (up to a null set). A measure-preserving map T : X → X is ergodic if every T -invariant measurable subset of X is either null or conull.
c XXXX American Mathematical Society
1 2 FRANKLIN, GREENBERG, MILLER, AND NG
Theorem 3 (Kuˇcera[7]). A sequence R ∈ 2ω is Martin-L¨ofrandom if and only if it is a Poincar´epoint for the shift operator with respect to the collection of effectively 0 ω closed (i.e., Π1) subsets of 2 . Building on work of Bienvenu, Day, Mezhirov and Shen [2], Bienvenu, Hoyrup and Shen generalised Kuˇcera’sresult to arbitrary computable ergodic transforma- tions of computable probability spaces. Theorem 4 (Bienvenu, Hoyrup and Shen [3]). Let (X, µ) be a computable proba- bility space, and let T : X → X be a computable ergodic transformation. A point x ∈ X is Martin-L¨ofrandom if and only if it is a Poincar´epoint for T with respect to the collection of effectively closed subsets of X.2 One of the most fundamental regularity theorems is due to Birkhoff (see [5]). Birkhoff’s Ergodic Theorem. Let (X, µ) be a probability space, and let T : X → X be ergodic. Let f ∈ L1(X). Then for almost all x ∈ X, 1 X Z lim f T i(x) = f dµ. n→∞ n i