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

• Autorzy: Vitaly Bergelson , Tomasz Downarowicz
• Języki publikacji: EN

Abstrakty

EN

We consider a hierarchy of notions of largeness for subsets of 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 βℤ to establish connections between various notions of largeness and apply those results to the study of the sets $R^{ε}_{A,B} = {n ∈ ℤ: μ(A ∩ TⁿB) > μ(A)μ(B) - ε}$ of times of "fat intersection". Among other things we show that the sets $R^{ε}_{A,B}$ 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 ℤ-actions.

Kategorie tematyczne

• 37A25: Ergodicity, mixing, rates of mixing
• 37B20: Notions of recurrence

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2008
• Tom: 110
• Numer: 1
• Strony: 117-150

Daty

wydano

2008

Twórcy

autor

Vitaly Bergelson

• Department of Mathematics, The Ohio State University, 100 Math Tower, 231 West 18th Avenue, Columbus, OH 43210-1174, U.S.A.

autor

Tomasz Downarowicz

• Institute of Mathematics, Technical University, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

Identyfikatory

DOI

10.4064/cm110-1-4