Weak Rational Ergodicity Does Not Imply Rational Ergodicity

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.

15 pages