and the ergodic decomposition

Mathieu Hoyrup

LORIA 615, rue du jardin botanique BP 239 54506 Vandoeuvre-l`es-Nancy France [email protected]

Abstract. We briefly present ongoing work about Martin-L¨ofrandomness and the ergodic decompo- sition theorem.

In [ML66], Martin-L¨ofdefined 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 no- tion has proved successful, as it turns out that random sequences in the sense of Martin-L¨of 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 the- orems 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 Section1 for definitions and details):

1. Let E be the set of points x such that Px is ergodic. E has one for every 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 A, µ(A) = R 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 non- constructive, or non-computable. We briefly survey a few investigations about this problem, especially from the point of view of Martin-L¨ofrandomness. 1 The ergodic decomposition

We work with a topological , i.e. a compact metric space X and a con- tinuous transformation T : X → X. B is the σ-field of Borel subsets of X. Let µ be a T -invariant Borel over X (i.e., µ(A) = µ(T −1A) for A ∈ B). Birkhoff’s ergodic theorem states that for every observable f ∈ L1(X, B, µ), the following limit exists

1 X f ∗(x) := lim f ◦ T i(x) for µ-almost every x ∈ X. (1) n→∞ n i

Moreover, the f ∗ is integrable and R f ∗ dµ = R f dµ. In general one cannot expect, for a single point x, convergence (1) for every f ∈ L1(X, B, µ). Nevertheless, using a separability argument it is easy to see that for µ-almost every x, con- vergence holds for every bounded continuous f. Let us denote the set of continuous functions from X to R by C(X). A point is called generic for T if f ∗(x) exists for all f ∈ C(X) (this notion does not depend on µ). Now, given a generic point x, the functional which maps f ∈ C(X) to f ∗(x) is positive and linear, so it extends to a probability measure, denoted P . In other words, x Z ∗ f (x) = f dPx

for every f ∈ C(X). ∗ From the definition of f (x), one can prove that for every generic x, Px is T -invariant. Now, the ergodic decomposition theorem states if µ is T -invariant then Px is ergodic for µ- almost every x. This theorem admits (at least) two different proofs, one is an application of Choquet’s theorem from convex analysis: the set of invariant measures is a compact convex set whose extreme points are the ergodic measures; hence, any invariant measure µ can be expressed as a barycenter of the ergodic measures, i.e. there is a probability measure m over the set of probability measures over X assigning full weight to the set E(T ) of ergodic invariant measures, such that for every Borel set A, Z µ(A) = ν(A) dm(ν). (2)

2 Martin-L¨ofrandom points

A natural way of investigating the constructivity of the ergodic decomposition is the fol- lowing: given a topological system with appropriate computability assumptions, given an invariant measure µ, is Px ergodic for every µ-ML random point x? Precisely we will assume X is a computable metric space that is compact in an effective way, T : X → X a computable map and µ a Borel probability measure over X that is T -invariant. Observe that it is a consequence of V’yugin’s theorem that if x is µ-ML random then x is generic, so Px is well-defined. It is important to note that while V’yugin’s theorem was originally proved when X is the Cantor space and µ is computable, it can be extended

2 to arbitrary computable metric spaces and arbitrary Borel probability measures [GHR], [Hoy08]. We briefly investigate the question: is Px ergodic?

Remark 1. First, if µ is itself ergodic, then from V’yugin’s theorem, Px = µ, so Px is ergodic.

2.1 Naive strategy ∗ Let x be a generic point and Fx be the class of Borel sets A such that 1A(x) = µ(A)(1A is ∗ the indicator of A and 1A is defined by (1)). All T -invariant sets A ∈ Fx have Px-measure 0 or 1, as the whole trajectory of x lies either inside A, or outside A. If Fx were the whole class of Borel sets (which is almost never the case) then Px would be automatically ergodic. Hence this approach (which does not work) raises two questions:

1. Find a characterization of Fx for µ-random x. For instance, does Fx contain all the r.e. open sets? We know that it contains all the r.e. open sets whose µ-measure is computable (see [HR09]). 2. Which classes F are sufficient to characterize ? A class F of Borel sets char- acterizes ergodicity if the system is ergodic as soon as all the T -invariant sets in F have measure 0 or 1. In particular, can we restrict to sets that are constructive in some sense?

2.2 Other strategy The following proposition is an easy consequence of classical results. Proposition 1. Every m-random measure is ergodic.

Proof. It is known that the set of ergodic measures is a Gδ-set. When (X,T ) is a computable system, it is a computable Gδ-set. As it has measure one, it contains all the random elements, by a result proved by Kurtz (actually, its relativized version for non-computable measures). We prove the following result: Theorem 1. Assume m and µ are computable. The following are equivalent: 1. x is µ-random, 2. there exists a m-random measure ν such that x is ν-random. Hence, if m is computable the problem is solved: if x is a µ-random point, then there is a m-random measure ν such that x is ν-random. By the preceding proposition, ν is ergodic, so by Remark1, Px = ν. The problem boils down to the computability of m and µ. As µ can be simply derived from m by equality (2), it follows that µ is always computable relative to m (this can be formalized using Weirauch’s type-two machines or Ko’s oracle machines for instance, see [Wei00], [Ko91]). But it might be possible that a computable measure µ induces a non- computable m. When m is not computable, a relativized version of Theorem1 gives that if x is µ-random relative to m, then there exists a m-random measure ν such that x is ν-random, so as above, ν is ergodic and ν = Px. But we do not know what happens for µ-random points that are no more random when m is added as an oracle.

3 3 Summary

We can identify three ways of defining the computability of the ergodic decomposition:

1. Given a generic point x, is Px computable from x? In what sense? – The point in V’yugin’s counter-example is that the mapping x 7→ Px is non-effective, in some sense. In particular, there is a computable observable f such that f ∗ is not L1-computable. 2. Is the measure m computable? – In V’yugin’s counter-example, even if x 7→ Px is non-effective, the measure m is computable. 3. Given a µ-random point x, is Px ergodic? For the moment we only know that 1. =⇒ 2. =⇒ 3., for a suitable notion of computability of the mapping x 7→ Px. In the introduction, we said that in V’yugin’s example the ergodic decomposition is non- constructive. Actually, it is computable in the sense of 2. (and hence 3.) but not in the sense of 1. We still do not know whether the ergodic decomposition is always computable in the sense of 2. (while we conjecture it is not the case) and 3.

References

AGT10. J. Avigad, P. Gerhardy, and H. Towsner. Local stability of ergodic averages. Trans. Amer. Math. Soc., 362:261–288, 2010.1 GHR. S. Galatolo, M. Hoyrup, and C. Rojas. Effective symbolic dynamics, random points, statistical behavior, complexity and entropy. Information and Computation. To appear.3 Hoy08. M. Hoyrup. Computability, Randomness and on metric spaces. PhD thesis, Universit´eDenis- Diderot – Paris VII, 2008.3 HR09. Mathieu Hoyrup et Crist´obalRojas. Applications of Effective Probability Theory to Martin-L¨ofRandom- ness. ICALP 2009. LNCS, 5555:549-561, 2009. 1,4 Ko91. K.-I Ko. Complexity Theory of Real Functions. Birkhauser Boston Inc., Cambridge, MA, USA, 1991.5 ML66. P. Martin-L¨of.The definition of random sequences. Information and Control, 9(6):602–619, 1966.1 Nan08. Satyadev Nandakumar. An effective ergodic theorem and some applications. In STOC ’08: Proceedings of the 40th annual ACM symposium on Theory of computing, pages 39–44, New York, NY, USA, 2008. ACM. 1 V’y97. V. V. V’yugin. Effective convergence in probability and an ergodic theorem for individual random sequences. SIAM Theory of Probability and Its Applications, 42(1):39–50, 1997.1 Wei00. K. Weihrauch. Computable Analysis. Springer, Berlin, 2000.5

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