Randomness and the Ergodic Decomposition

We briefly present ongoing work about Martin-Löf randomness and the ergodic decomposition theorem. In [ML66], Martin-Löf defined the notion of an algorithmically random infinite binary sequence. The idea was to have an individual notion of a random object, an object that is acceptable as an outcome of a probabilistic process (typically, the tossing of a coin). This notion has proved successful, as it turns out that random sequences in the sense of Martin-Löf have all the usual properties that are proved to hold almost surely in classical probability theory. A particularly interesting property is typicalness in the sense of Birkhoff’s ergodic theorem. This theorem embodies many probability theorems (strong law of large numbers, diophantine approximation, e.g.). Whereas the algorithmic versions of many probability theorems are straightforward to derive from their classical proofs, whether the ergodic theorem has a version for random elements has been an open problem for years, finally proved by V’yugin [V’y97] (improvements of this result, extending the class of functions for which the algorithmic version holds, have been established later in [Nan08], [HR09]). The reason for this difficulty is that the classical proof of Birkhoff’s theorem is in some sense nonconstructive. In [V’y97], V’yugin gave it a precise meaning: the speed of convergence is not computable in general. Recently, Avigad, Gerhardy and Towsner [AGT10] proved that the speed of convergence is actually computable in the ergodic case, i.e. when the system is undecomposable. As a result, the non-constructivity of the theorem lies in the ergodic decomposition. Let us recall two ways of talking about the ergodic decomposition (see Section 1 for definitions and details): 1. Let E be the set of points x such that Px is ergodic. E has measure one for every invariant measure. 2. For each invariant measure μ, there is a measure m supported on the ergodic measures, such that μ is the barycenter of m, i.e. for every Borel set A, μ(A) = ∫ ν(A) dm(ν). One can easily derive each formulation from the other, but not constructively. We are interested in the precise extent to which the ergodic decomposition is nonconstructive, or non-computable. We briefly survey a few investigations about this problem, especially from the point of view of Martin-Löf randomness. 1 The ergodic decomposition We work with a topological dynamical system, i.e. a compact metric space X and a continuous transformation T : X → X. B is the σ-field of Borel subsets of X. Let μ be a T -invariant Borel probability measure over X (i.e., μ(A) = μ(T−1A) for A ∈ B). Birkhoff’s ergodic theorem states that for every observable f ∈ L(X,B, μ), the following limit exists