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A good universal weight for nonconventional ergodic averages in norm

Published online by Cambridge University Press:  28 December 2015

IDRIS ASSANI  and
RYO MOORE
IDRIS ASSANI
Affiliation:
Department of Mathematics, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA email assani@math.unc.edu, ryom@live.unc.edu
RYO MOORE
Affiliation:
Department of Mathematics, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA email assani@math.unc.edu, ryom@live.unc.edu
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Abstract

We will show that the sequence appearing in the double recurrence theorem is a good universal weight for the Furstenberg averages. That is, given a system $(X,{\mathcal{F}},\unicode[STIX]{x1D707},T)$ and bounded functions $f_{1},f_{2}\in L^{\infty }(\unicode[STIX]{x1D707})$, there exists a set of full-measure $X_{f_{1},f_{2}}$ in $X$ that is independent of integers $a$ and $b$ and a positive integer $k$ such that, for all $x\in X_{f_{1},f_{2}}$, for every other measure-preserving system $(Y,{\mathcal{G}},\unicode[STIX]{x1D708},S)$ and for each bounded and measurable function $g_{1},\ldots ,g_{k}\in L^{\infty }(\unicode[STIX]{x1D708})$, the averages

$$\begin{eqnarray}\frac{1}{N}\mathop{\sum }_{n=1}^{N}f_{1}(T^{an}x)f_{2}(T^{bn}x)g_{1}\circ S^{n}g_{2}\circ S^{2n}\cdots g_{k}\circ S^{kn}\end{eqnarray}$$
converge in $L^{2}(\unicode[STIX]{x1D708})$.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 4 , June 2017 , pp. 1009 - 1025
Copyright
© Cambridge University Press, 2015 

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