Effective symbolic dynamics, random points, statistical behavior, complexity and entropy

Martin-Löf’s original definition was given for infinite symbolic sequences. With this notion each single random sequence behaves as a generic sequence of the probability space for each effective statistical test. In this way many probabilistic theorems having almost everywhere statements can be translated to statements which hold for each random sequence. As an example we cite the fact that in each infinite string of 0’s and 1’s which is random for the uniform measure, all the digits appear with the same limit frequency. This is a particular case, related to the strong law of large numbers (or Birkhoff ergodic theorem). A general statement of this kind was given by V’yugin (Birkhoff ergodic theorem for individual random sequences, see [V’y97] and lemma 3.2.1 below).

Recently the notion of Martin-Löf randomness was generalized to computable metric spaces endowed with a measure ( [Gác05,HR07]). Computable metric spaces are separable metric spaces where the distance can be in some sense effectively computed (see section 2.4 ). In those spaces, it is also possible to define “computable” functions, which are functions whose behavior is in some sense given by an algorithm, and “computable” measures (there is an algorithm to calculate the measure of nice sets). The space of infinite symbolic sequences, the real line or euclidean spaces, are examples of metric spaces which become computable in a very natural way.

A particularly interesting class of general stationary stochastic processes is constituted by those generated by a measure-preserving map on a metric space, these are the objects studied by ergodic theory. In this paper we consider systems of the type (X, T, µ), where X is a computable metric space, µ a computable probability measure and T a computable endomorphism. The above considered symbolic shifts on spaces of infinite sequences which preserve a computable measure are systems of this kind.

In the classical ergodic theory, a powerful technique (symbolic dynamics) allows to associate to a general system as above (X, T, µ) a shift on a space of infinite strings having similar statistical properties. In section 3 we use the algorithmic features of computable metric spaces to define computable measurable partitions and construct effective symbolic models for the dynamics. In this models random points are associated to random infinite strings. This tool allows to generalize theorems which are proved in the symbolic setting to the more general setting of maps and metric spaces. For example the above cited V’yugin theorem becomes a Birkhoff theorem for random points. On this line, we also prove a Poincaré recurrence theorem for random points. Those statements (see thm.3.2.1 and prop.

Theorem. Let (X, µ) be a computable probability space. If x is µ-random, then it is recurrent with respect to every measure preserving endomorphism T on (X, µ).

Moreover, each µ-random point x is typical for every ergodic endomorphism T , i.e.

for every continuous bounded real-valued f .

In the remaining part of the paper these tools are also used to prove relations between various definitions of orbit complexity and entropy of the systems.

In [Bru83], Brudno defined a notion of algorithmic complexity K(x, T ) for the orbits of a dynamical system on a compact space. It is a measure of the information rate which is necessary to describe the behavior of the orbit of x. In this pointwise definition the information is measured by the Kolmogorov information content. Later, White ([Whi93]) also introduced a slightly different version K(x, T ). Brudno then proved the following results, later improved by White: Theorem (Brudno, White). Let X be a compact topological space and T : X → X a continuous map.

(1) For any ergodic probability measure µ the equality K(x, T ) = K(x, T ) = h µ (T ) holds for µ-almost all x ∈ X, (2) For all x ∈ X, K(x, T ) ≤ h(T ).

where h µ (T ) is the Kolmogorov-Sinaï entropy of (X, T ) with respect to µ and h(T ) is the topological entropy of (X, T ). This result seems miraculous as no computability assumption is required on the space or on the transformation T . Actually, this miracle lies in the compactness of the space, which makes it finite when observations are made with finite precision (open covers of the space can be reduced to finite open covers). Indeed, when the space is not compact, it is possible to construct systems for which the algorithmic complexity of orbits is correlated in no way to their dynamical complexity. In [Gal00], Brudno’s definition was generalized to non-compact computable metric spaces. This definition coincides with

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