Ergodic Properties of Square-Free Numbers
  We construct a natural invariant measure concentrated on the set of square-free numbers, and invariant under the shift. We prove that the corresponding dynamical system is isomorphic to a translation on a compact, Abelian group. This implies that this system is not weakly mixing and has zero measure-theoretical entropy. 
Francesco Cellarosi, Yakov G. Sinai
math.DS math.NT