Ergodic properties of square-free numbers

F. Cellarosi; Ya G. Sinai

doi:10.4171/JEMS/394

Ergodic properties of square-free numbers

F. Cellarosi, Ya G. Sinai

Research output: Contribution to journal › Article › peer-review

36 Scopus citations

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.

Original language	English (US)
Pages (from-to)	1343-1374
Number of pages	32
Journal	Journal of the European Mathematical Society
Volume	15
Issue number	4
DOIs	https://doi.org/10.4171/JEMS/394
State	Published - 2013

All Science Journal Classification (ASJC) codes

General Mathematics
Applied Mathematics

Keywords

Correlation functions
Dynamical systems with pure point spectrum
Ergodicity
Square-free numbers

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10.4171/JEMS/394

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title = "Ergodic properties of square-free numbers",

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.",

keywords = "Correlation functions, Dynamical systems with pure point spectrum, Ergodicity, Square-free numbers",

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T1 - Ergodic properties of square-free numbers

AU - Cellarosi, F.

AU - Sinai, Ya G.

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

KW - Correlation functions

KW - Dynamical systems with pure point spectrum

KW - Ergodicity

KW - Square-free numbers

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JO - Journal of the European Mathematical Society

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Ergodic properties of square-free numbers

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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