## Illinois Journal of Mathematics

### Convergence of polynomial ergodic averages of several variables for some commuting transformations

Michael C. R. Johnson

#### Abstract

Let $(X,\mathcal{B},\mu)$ be a probability space and let $T_1,\ldots , T_l$ be $l$ commuting invertible measure preserving transformations of $X$. We show that if $T_1^{c_1} \ldots T_l^{c_l}$ is ergodic for each $(c_1,\ldots ,c_l)\neq(0,\ldots,0)$, then the averages $\frac{1}{|\Phi_N|}\sum_{u\in\Phi_N}\prod _{i=1}^r T_1^{p_{i1}(u)}\ldots T_l^{p_{il}(u)}f_i$ converge in $L^2(\mu)$ for all polynomials $p_{ij} : \mathbb {Z}^d\to\mathbb{Z}$, all $f_i\in L^{\infty}(\mu)$, and all Følner sequences $\{\Phi_N\}_{N=1}^{\infty}$ in $\mathbb{Z}^d$.

#### Article information

Source
Illinois J. Math., Volume 53, Number 3 (2009), 865-882.

Dates
First available in Project Euclid: 4 October 2010

https://projecteuclid.org/euclid.ijm/1286212920

Digital Object Identifier
doi:10.1215/ijm/1286212920

Mathematical Reviews number (MathSciNet)
MR2727359

Zentralblatt MATH identifier
1210.28017

#### Citation

Johnson, Michael C. R. Convergence of polynomial ergodic averages of several variables for some commuting transformations. Illinois J. Math. 53 (2009), no. 3, 865--882. doi:10.1215/ijm/1286212920. https://projecteuclid.org/euclid.ijm/1286212920

#### References

• V. Bergelson, Weakly mixing PET, Ergodic Theory Dynam. Systems 7 (1987), 337–349.
• V. Bergelson and A. Leibman, Polynomial extensions of van der Waerden's and Szemerédi's theorems, J. Amer. Math. Soc. 9 (1996), 725–753.
• V. Bergelson, R. McCutcheon and Q. Zhang, A Roth theorem for amenable groups, Amer. J. Math. 119 (1997), 1173–1211.
• N. Frantzikinakis and B. Kra, Convergence of multiple ergodic averages for some commuting transformations, Ergodic Theory Dynam. Systems 25 (2005), 799–809.
• H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Anal. Math. 31 (1977), 204–256.
• B. Host and B. Kra, Nonconventional ergodic averages and nilmanifolds, Ann. of Math. (2) 161 (2005), 397–488.
• B. Host and B. Kra, Convergence of polynomial ergodic averages, Israel J. Math. 149 (2005), 1–19.
• A. Leibman, Pointwise convergence of ergodic averages for polynomial actions of $\mathbb{Z}^d$ by translation on a nilmanifold, Ergodic Theory Dynam. Systems 25 (2005), 215–225.
• A. Leibman, Convergence of multiple ergodic averages along polynomials of several variables, Israel J. Math. 146 (2005), 303–315.
• T. Tao, Norm convergence of multiple ergodic averages for commuting transformaions, Ergodic Theory Dynam. Systems 10 (2008), 657–688.
• N. Wiener, The ergodic theorem, Duke Math. J. 5 (1939), 1–18.