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Summer 2011 Multiple ergodic averages for flows and an application
Amanda Potts
Illinois J. Math. 55(2): 589-621 (Summer 2011). DOI: 10.1215/ijm/1359762404


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.


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Amanda Potts. "Multiple ergodic averages for flows and an application." Illinois J. Math. 55 (2) 589 - 621, Summer 2011.


Published: Summer 2011
First available in Project Euclid: 1 February 2013

zbMATH: 1300.37003
MathSciNet: MR2941317
Digital Object Identifier: 10.1215/ijm/1359762404

Primary: 37A10
Secondary: 05D10, 11B05

Rights: Copyright © 2011 University of Illinois at Urbana-Champaign


Vol.55 • No. 2 • Summer 2011
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