Rational weak mixing in infinite measure spaces

TL;DR

This paper explores rational weak mixing in infinite measure spaces, establishing its properties, implications, and genericity, with applications to Markov shifts exhibiting strong ratio limit properties.

Contribution

It introduces and analyzes rational weak mixing in infinite measure spaces, connecting it to weak ergodicity, weak mixing, and demonstrating its genericity.

Findings

Rational weak mixing implies weak rational ergodicity and weak spectral mixing.

Markov shifts with Orey's strong ratio limit property exhibit rational weak mixing.

The power, subsequence version of rational weak mixing is generic.

Abstract

Rational weak mixing is a measure theoretic version of Krickeberg's strong ratio mixing property for infinite measure preserving transformations. It requires "{\tt density}" ratio convergence for every pair of measurable sets in a dense hereditary ring. Rational weak mixing implies weak rational ergodicity and (spectral) weak mixing. It is enjoyed for example by Markov shifts with Orey's strong ratio limit property. The power, subsequence version of the property is generic.