Illinois Journal of Mathematics

Multiple ergodic averages for flows and an application

Amanda Potts

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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

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

First available in Project Euclid: 1 February 2013

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Zentralblatt MATH identifier

Primary: 37A10: One-parameter continuous families of measure-preserving transformations
Secondary: 11B05: Density, gaps, topology 05D10: Ramsey theory [See also 05C55]


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

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