Empirical Approach to the ×2, ×3 Conjecture

Abstract

We study atomic measures on $[0,1]$ which are invariant both under multiplication by $2\hspace{1em}\mathrm{mod}1$ and by $3\hspace{1em}\mathrm{mod}1,$ since such measures play an important role in deciding Furstenberg’s $\times 2,\times 3$ conjecture. Our specific focus was finding atomic measures whose supports are far from being uniformly distributed, and we used computer software to discover a number of such measures (which we call outlier measures). The structure of these measures indicates the possibility that a sequence of atomic measures may converge to a non-Lebesgue measure; likely one which is a combination of the Lebesgue measure and one or more atomic measures.

Keywords:

• Furstenberg conjecture
• dynamical system
• ergodic measure

Acknowledgments

Research of both authors is supported from:

• Resources for science in years 2013–2018 as research project (NCN grant 2013/08/A/ST1/00275, Poland

• Statutory research funds of Faculty of Pure and Applied Mathematics at Wrocław University of Science and Technology

Notes

1 The original conjecture deals more generally with $T(x)=\mathit{px}\hspace{1em}\mathrm{mod}1$ and $S(x)=\mathit{qx}\hspace{1em}\mathrm{mod}1$ with p, q being a pair of coprime natural numbers; the pair (2, 3) is the simplest such pair.

Additional information

Funding

This study was supported by Narodowe Centrum Nauki (2013/08/A/ST1/00275) and Politechnika Wrocławska (0401/0155/18).