Abstract
We study atomic measures on which are invariant both under multiplication by and by since such measures play an important role in deciding Furstenberg’s 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.
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 and with p, q being a pair of coprime natural numbers; the pair (2, 3) is the simplest such pair.