Randomness on Computable Probability Spaces - A Dynamical Point of View

TL;DR

This paper extends the concept of Schnorr randomness to computable probability spaces and establishes its equivalence to typicality in mixing computable dynamical systems, developing new tools for the theory.

Contribution

It introduces a new framework linking Schnorr randomness with dynamical typicality in computable probability spaces and develops tools like morphisms for this theory.

Findings

Schnorr randomness equals typicality for all mixing computable dynamics.

Developed new tools for computable probability spaces, such as morphisms.

Established a characterization connecting randomness and dynamical behavior.

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.