Non-rigid rank-one infinite measures on the circle

Abstract

For a class of irrational numbers, depending on their Diophantine properties, we construct explicit rank-one transformations that are totally ergodic and not weakly mixing and classify when the measure is finite or infinite. In the finite case they are isomorphic to irrational rotations, giving explicit rank-one cutting and spacer parameters for these irrational rotations. In the infinite case we use the constructions to provide examples of non-weakly mixing infinite measure-preserving ergodic transformations which do not have any nontrivial probability preserving factors with discrete spectrum, thereby contributing to the program of Aaronson and Nadkarni and a question of Glasner and Weiss. We also obtain nonsingular versions of these examples for each Krieger ration set.