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

Johnson, Michael C. R.

Illinois J. Math., Tome 53 (2009) no. 1, p. 865-882 / Harvested from Project Euclid

Résumé

Let $(X,\mathcal{B},\mu)$ be a probability space and let T₁, …, T_l be l commuting invertible measure preserving transformations of X. We show that if T₁^c₁…T_l^c_l is ergodic for each (c₁, …, c_l)≠(0, …, 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 L²(μ) for all polynomials p_ij : ℤ^d→ℤ, all f_i∈L^∞(μ), and all Følner sequences {Φ_N}_N=1^∞ in ℤ^d.

Publié le : 2009-05-15
Classification: 28D05, 37A15

@article{1286212920,
     author = {Johnson, Michael C. R.},
     title = {Convergence of polynomial ergodic averages of several variables for some commuting transformations},
     journal = {Illinois J. Math.},
     volume = {53},
     number = {1},
     year = {2009},
     pages = { 865-882},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1286212920}
}

Johnson, Michael C. R. Convergence of polynomial ergodic averages of several variables for some commuting transformations. Illinois J. Math., Tome 53 (2009) no. 1, pp.  865-882. http://gdmltest.u-ga.fr/item/1286212920/