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Martin-Löf random points satisfy Birkhoff’s ergodic theorem for effectively closed sets
| Content Provider | Semantic Scholar |
|---|---|
| Author | Franklin, Johanna N. Y. Greenberg, Noam Miller, Joseph S. Ng, Keng Meng |
| Copyright Year | 2012 |
| 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 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, 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(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 Kučera’s classic result characterising MartinLöf randomness in the Cantor space. We reformulate Kučera’s result 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 Poincaré point for T with respect to C if for all E ∈ C of positive measure for infinitely many n, T(x) ∈ E. The Cantor space 2 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 (2, λ). Received by the editors July 20, 2010 and, in revised form, April 5, 2011 and April 8, 2011. 2010 Mathematics Subject Classification. Primary 03D22; Secondary 28D05, 37A30. 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 DMS0945187 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 ©2012 American Mathematical Society Reverts to public domain 28 years from publication 3623 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 3624 J.N.Y. FRANKLIN, N. GREENBERG, J.S. MILLER, AND K.M. NG Theorem 3 (Kučera [7]). A sequence R ∈ 2 is Martin-Löf random if and only if it is a Poincaré point for the shift operator with respect to the collection of effectively closed (i.e., Π1) subsets of 2 . Building on work of Bienvenu, Day, Mezhirov and Shen [2], Bienvenu, Hoyrup and Shen generalised Kučera’s result to arbitrary computable ergodic transformations of computable probability spaces. Theorem 4 (Bienvenu, Hoyrup and Shen [3]). Let (X,μ) be a computable probability space, and let T : X → X be a computable ergodic transformation. A point x ∈ X is Martin-Löf random if and only if it is a Poincaré point for T with respect to the collection of effectively closed subsets of X. 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 ∈ L(X). Then for almost all x ∈ X, |
| Starting Page | 3623 |
| Ending Page | 3628 |
| Page Count | 6 |
| File Format | PDF HTM / HTML |
| DOI | 10.1090/S0002-9939-2012-11179-7 |
| Volume Number | 140 |
| Alternate Webpage(s) | http://www.ntu.edu.sg/home/kmng/Files/Papers/birkhoff.pdf |
| Alternate Webpage(s) | http://www.math.wisc.edu/~jmiller/Papers/birkhoff.pdf |
| Alternate Webpage(s) | http://www.ams.org/journals/proc/2012-140-10/S0002-9939-2012-11179-7/S0002-9939-2012-11179-7.pdf |
| Alternate Webpage(s) | http://homepages.mcs.vuw.ac.nz/~greenberg/Papers/30-birkhoff.pdf |
| Alternate Webpage(s) | https://doi.org/10.1090/S0002-9939-2012-11179-7 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
