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.
All Science Journal Classification (ASJC) codes
- Applied Mathematics
- Correlation functions
- Dynamical systems with pure point spectrum
- Square-free numbers