Computing the speed of convergence of ergodic averages and pseudorandom points in computable dynamical systems

TL;DR

This paper provides a simpler proof that in computable ergodic systems, pseudorandom points with computable dynamics exhibit effective convergence of ergodic averages, and if the invariant measure is also computable, these points are densely distributed.

Contribution

The paper offers an alternative, simplified proof of effective convergence of ergodic averages in computable systems and establishes the density of pseudorandom points when the invariant measure is computable.

Findings

Effective convergence of ergodic averages proven with a new simpler method

Pseudorandom points form a dense set in the support of the invariant measure

Density of pseudorandom points when the invariant measure is computable

Abstract

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. It was proved in [Avigad et al. 2010, Local stability of ergodic averages] that in a system whose dynamics is computable the ergodic averages of computable observables converge effectively. We give an alternative, simpler proof of this result. This implies that if also the invariant measure is computable then the pseudorandom points are a set which is dense (hence nonempty) on the support of the invariant measure.