Convergence of multiple ergodic averages along polynomials of several variables
Let \(T\) be an invertible measure-preserving transformation on the probability measure space \((X, \mathcal{B}, \mu).\) The author generalizes an earlier result of \textit{B. Host} and \textit{B. Kra} [Convergence of polynomial ergodic averages. Isr. J. Math. 149, 1--19 (2005)]. He proves that for any natural numbers \(d, r,\) for any polynomials \(p_k, 1\leq k \leq r\), convergence of polynomial ergodic averages. from \(\mathbb Z^d\) to \(\mathbb Z\) and for any Folner sequence \(\Phi_N\) in \(\mathbb{Z}^d\) the averages \[ {1\over {| \Phi_N| }}\sum_{u\in \Phi_N}T^{p_1(u)}f_1\dots T^{p_r(u)}f_r \] converge in \(L^2\) norm for all bounded functions \(f_j, 1\leq j\leq r.\) The main ingredients in the proof are the following: 1) The seminorms introduced by \textit{B. Host} and \textit{B. Kra} [Ann. Math. (2) 161, 397-488 (2005; Zbl 1077.37002)] and the factors they characterize on which \(T\) acts as a translation. 2) A pointwise convergence result on these factors due to the author [Ergodic Theory Dyn. Syst. 25, 215--225 (2005; Zbl 1080.37004)] from which the \(L^2\) convergence is derived without difficulty. 3) An induction similar to the one used by \textit{V. Bergelson} [Ergodic Theory Dyn. Syst. 7, 337--349 (1987; Zbl 0645.28012)] to move from polynomials of degree one to those with arbitrary degree.
scientific article; zbMATH DE number 2174714