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Pointwise multiple averages for systems with two commuting transformations

Published online by Cambridge University Press:  14 March 2017

SEBASTIÁN DONOSO  and
WENBO SUN Open the ORCID record for WENBO SUN [Opens in a new window]
SEBASTIÁN DONOSO
Affiliation:
Center for Mathematical Modeling, University of Chile, Beauchef 851, Santiago, Chile email sdonoso@dim.uchile.cl
WENBO SUN
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road Evanston, IL 60208-2730, USA email swenbo@math.northwestern.edu
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Abstract

We show that for every ergodic measure-preserving system $(X,{\mathcal{X}},\unicode[STIX]{x1D707},S,T)$ with commuting transformations $S$ and $T$, the average

$$\begin{eqnarray}\frac{1}{N^{3}}\mathop{\sum }_{i,j,k=0}^{N-1}f_{0}(S^{j}T^{k}x)f_{1}(S^{i+j}T^{k}x)f_{2}(S^{j}T^{i+k}x)\end{eqnarray}$$
converges for $\unicode[STIX]{x1D707}$-almost every $x\in X$ as $N\rightarrow \infty$ for all $f_{0},f_{1},f_{2}\in L^{\infty }(\unicode[STIX]{x1D707})$. We also show that if $(X,{\mathcal{X}},\unicode[STIX]{x1D707},S,T)$ is an ergodic measurable distal system, then the average
$$\begin{eqnarray}\frac{1}{N}\mathop{\sum }_{i=0}^{N-1}f_{1}(S^{i}x)f_{2}(T^{i}x)\end{eqnarray}$$
 converges for $\unicode[STIX]{x1D707}$-almost every $x\in X$ as $N\rightarrow \infty$ for all $f_{1},f_{2}\in L^{\infty }(\unicode[STIX]{x1D707})$.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 6 , September 2018 , pp. 2132 - 2157
Copyright
© Cambridge University Press, 2017 

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