Randomness on Computable Probability Spaces—A Dynamical Point of View

Abstract

Abstract. We extend the notion of randomness (in the version introduced by Schnorr) to computable Probability Spaces and compare it to a dynamical notion of randomness: typicality. Roughly, a point is typical for some dynamic, if it follows the statistical behavior of the system (Birkhoff's pointwise ergodic theorem). We prove that a point is Schnorr random if and only if it is typical for every mixing computable dynamics. To prove the result we develop some tools for the theory of computable probability spaces (for example, morphisms) that are expected to have other applications.

Schnorr randomness and the Lebesgue differentiation theorem

“…In [7], a characterization of Schnorr randomness in terms of dynamical typicalness was given. Here we state a slightly improved version, obtained using a result from [1] (see also [8]) which concerns the computability of the rate of convergence of ergodic averages.…”

Fermat’s last theorem and chaoticity

“…Recently, [9] showed that in classical chaotic dynamics, typicality corresponds exactly to Schnorr randomness; this means that a chaotic system may produce a computable sequence of bits provided the initial point is suitable chosen, but this event has probability zero (the set of initial points can be infinite).…”

Martin-Löf random points satisfy Birkhoff’s ergodic theorem for effectively closed sets

“…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.…”

“…2 Here we use the notion of a computable probability space of Gács, Hoyrup and Rojas [4], although all reasonable definitions of this concept are equivalent. For completeness, we recall the definition here.…”

“…A computable probability space is a computable metric space equipped with a computable measure. Gács, Hoyrup and Rojas [4] characterised the Schnorr random points as the Birkhoff points for computable ergodic transformations with respect to effectively closed sets whose measure is computable. They asked [8] what happens if we omit the requirement that the measure of the sets be computable.…”