On ergodic transformations that are both weakly mixing and uniformly   rigid
  We examine some of the properties of uniformly rigid transformations, and analyze the compatibility of uniform rigidity and (measurable) weak mixing along with some of their asymptotic convergence properties. We show that on Cantor space, there does not exist a finite measure-preserving, totally ergodic, uniformly rigid transformation. We briefly discuss general group actions and show that (measurable) weak mixing and uniform rigidity can coexist in a more general setting. 
Jennifer James, Thomas Koberda, Kathryn Lindsey, Cesar E. Silva, Peter   Speh