On rationally ergodic and rationally weakly mixing rank-one transformations

Abstract

Abstract. We study the notions of weak rational ergodicity and rational weak mixing as defined by Jon Aaronson. We prove that various families of infinite measure-preserving rank-one transformations possess (or do not posses) these properties, and consider their relation to other notions of mixing in infinite measure.

Strict doubly ergodic infinite transformations

“…For t > 1/2, T × T is not is not conservative. By [12], rational weak mixing implies weak doubly ergodic, so this give another example of a double ergodic transformation with nonconservative cartesian square. Our examples below can also be chosen to be rigid (Theorem 6.5).…”

“…Now suppose that p > 1. By (12), |I p | < 4qhn−1 jp−ip . Also, for any z ∈ Z with −i p ≤ z < r n − j p , we may write…”

“…The proof of the lemma is trivial in the case where S n = ∅, so suppose first that I 1 is nonempty and p = 1. The bounds in (12) imply that |S n | = |I 1 | < 4qhn−1 j1−i1 ≤ 4qh n−1 . But by equation (6), r n > 8qγ n = 16qh n−1 .…”

Subsequence bounded rational ergodicity of rank-one transformations

“…For the genericity result, rather than working with the cutting and stacking construction definition of rank-one transformations (used in examples and in e.g. [11]), we work with the abstract definition of rank-one. As Aaronson [4, §10] has shown that the class of weakly rationally ergodic transformations is meager in the group of measurepreserving transformations, a consequence of our results is that there exist rank-one transformations that are subsequence boundedly rationally ergodic but not weakly rationally ergodic.…”

Ergodicity and conservativity of products of infinite transformations and their inverses

“…The methods that we use are combinatorial and probabilistic in nature. Propositions 2.3 and 2.4 use the notion of descendants, as introduced in [8], to turn the dynamics of the rank-one system into combinatorial characterizations.…”