Weak Rational Ergodicity Does Not Imply Rational Ergodicity
Abstract
We extend the notion of rational ergodicity to β-rational ergodicity for β > 1. Given β ∈ 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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