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2013•Journal of the European Mathematical SocietyOpen access

Ergodic properties of square-free numbers

Francesco Cellarosi, Yakov G. Sinai

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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.

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What this paper is about

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.

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Available 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.

Key concepts: Mathematics, Ergodic theory, Invariant measure, Abelian group, Square-free integer, Invariant (physics), Pure mathematics, Entropy (arrow of time)

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