Ergodic Properties of Square-Free Numbers

TL;DR

This paper constructs an invariant measure on square-free numbers, showing the associated dynamical system is isomorphic to a translation on a compact Abelian group, with implications for its mixing properties and entropy.

Contribution

It introduces a natural invariant measure on square-free numbers and proves the system's isomorphism to a translation on a compact Abelian group, revealing its ergodic properties.

Findings

The system is not weakly mixing.

It has zero measure-theoretical entropy.

The system is isomorphic to a translation on a compact Abelian group.

Abstract

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.