Large sets of integers and hierarchy of mixing properties of measure preserving systems

Vitaly Bergelson¹ ; Tomasz Downarowicz²

¹Department of Mathematics The Ohio State University 100 Math Tower 231 West 18th Avenue Columbus, OH 43210-1174, U.S.A.
²Institute of Mathematics Technical University Wybrzeże Wyspiańskiego 27 50-370 Wrocław, Poland

Colloquium Mathematicum, Tome 110 (2008) no. 1, pp. 117-150

Résumé

We consider a hierarchy of notions of largeness for subsets of ${\mathbb Z}$ (such as thick sets, syndetic sets, IP-sets, etc., as well as some new classes) and study them in conjunction with recurrence in topological dynamics and ergodic theory. We use topological dynamics and topological algebra in $\beta{\mathbb Z}$ to establish connections between various notions of largeness and apply those results to the study of the sets $R^\varepsilon_{A,B} = \{n\in{\mathbb Z}: \mu(A\cap T^nB)>\mu(A)\mu(B) - \varepsilon\}$ of times of “fat intersection”. Among other things we show that the sets $R^\varepsilon_{A,B}$ allow one to distinguish between various notions of mixing and introduce an interesting class of weakly but not mildly mixing systems. Some of our results on fat intersections are established in a more general context of unitary ${\mathbb Z}$-actions.

Détail

DOI : 10.4064/cm110-1-4

Keywords: consider hierarchy notions largeness subsets mathbb thick sets syndetic sets ip sets etc classes study conjunction recurrence topological dynamics ergodic theory topological dynamics topological algebra beta mathbb establish connections between various notions largeness apply those results study sets varepsilon mathbb cap varepsilon times fat intersection among other things sets varepsilon allow distinguish between various notions mixing introduce interesting class weakly mildly mixing systems results fat intersections established general context unitary mathbb actions