Non-Rigid Measures Non-Rigid Rank-One Infinite Measures on the Circle.

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. We classify when the measure is finite or infinite. In the finite case they are isomorphic to irrational rotations. We also obtain rank-one nonrigid infinite invariant measures for irrational rotations, and, for each Krieger type, nonsingular measures on irrational rotations.