Weak Rational Ergodicity Does Not Imply Rational Ergodicity

TL;DR

This paper introduces a new concept called $eta$-rational ergodicity and demonstrates through constructed examples that weak rational ergodicity does not necessarily imply rational ergodicity, answering an open question in ergodic theory.

Contribution

The paper extends rational ergodicity to $eta$-rational ergodicity and provides examples showing weak rational ergodicity does not imply rational ergodicity.

Findings

Constructed uncountable family of transformations

Showed weak rational ergodicity does not imply 2-rational ergodicity

Answered an open question in ergodic theory

Abstract

We extend the notion of rational ergodicity to $\beta$-rational ergodicity for $\beta > 1$. Given $\beta \in \mathbb R$ such that $\beta > 1$, we construct an uncountable family of rank-one infinite measure preserving transformations that are weakly rationally ergodic, but are not $\beta$-rationally ergodic. The established notion of rational ergodicity corresponds to 2-rational ergodicity. Thus, this paper answers an open question by showing that weak rational ergodicity does not imply rational ergodicity.