Weak Rational Ergodicity Does Not Imply Rational Ergodicity

We extend the notion of rational ergodicity to $β$-rational ergodicity for $β> 1$. Given $β\in \mathbb R$ such that $β> 1$, we construct an uncountable family of rank-one infinite measure preserving transformations that are weakly rationally ergodic, but are not $β$-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.

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