Illinois Journal of Mathematics

Multiple ergodic averages for flows and an application

Amanda Potts

Abstract

We show the $L^2$-convergence of continuous time ergodic averages of a product of functions evaluated at return times along polynomials. These averages are the continuous time version of the averages appearing in Furstenberg's proof of Szemerédi’s Theorem. For each average, we show that it is sufficient to prove convergence on special factors, the Host-Kra factors, which have the structure of a nilmanifold. We also give a description of the limit. In particular, if the polynomials are independent over the real numbers then the limit is the product of the integrals. We further show that if the collection of polynomials has “low complexity”, then for every set $E$ of real numbers with positive density and for every $\delta>0$, the set of polynomial return times for the “$\delta$-thickened” set $E_{\delta}$ has bounded gaps. We give bounds for the flow average complexity and show that in some cases the flow average complexity is strictly less than the discrete average complexity.

Article information

Source
Illinois J. Math., Volume 55, Number 2 (2011), 589-621.

Dates
First available in Project Euclid: 1 February 2013

https://projecteuclid.org/euclid.ijm/1359762404

Digital Object Identifier
doi:10.1215/ijm/1359762404

Mathematical Reviews number (MathSciNet)
MR2941317

Zentralblatt MATH identifier
1300.37003

Citation

Potts, Amanda. Multiple ergodic averages for flows and an application. Illinois J. Math. 55 (2011), no. 2, 589--621. doi:10.1215/ijm/1359762404. https://projecteuclid.org/euclid.ijm/1359762404

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