We extend Schnorr randomness (in the version introduced by Schnorr) to computable probability spaces and compare it to a dynamical notion of randomness: typicality.Expand

We pursue the study of the framework of layerwise computability introduced in a preceding paper and give three applications to Martin-Lof randomness.Expand

This paper offers some new results on randomness with respect to classes of measures, along with a didactic exposition of their context based on results that appeared elsewhere.Expand

We introduce and study the framework of layerwise computability which lies on Martin-Lof randomness and the existence of a universal randomness test.Expand

We prove the effective version of [email protected]s ergodic theorem for Martin-Lof random points and effectively open sets, improving the results previously obtained in this direction (in particular those of Vyugin and Hoyrup, Rojas).Expand

A pseudorandom point in an ergodic dynamical system over a computable metric space is a point which is computable but its dynamics has the same statistical behavior as a typical point of the system.Expand

In the general context of computable metric spaces and computable measures we prove a kind of constructive Borel-Cantelli lemma: given a sequence (constructive in some way) of sets A" i with effectively summable measures, there are computable points which are not contained in infinitely many A"i.Expand

We introduce and compare some notions of complexity for orbits in dynamical systems and prove: (i) that the complexity of the orbits of random points equals the Kolmogorov-Sinai entropy of the system.Expand