In this talk we discuss the pointwise convergence. By building some suitable strictly ergodic models, we prove that for an ergodic system $(X,\mathcal{X},\mu, T)$, $d\in\N$, $f_1, \ldots, f_d \in L^{\infty}(\mu)$, the averages
$$\frac{1}{N^2} \sum_{(n,m)\in [0,N-1]^2} f_1(T^nx)f_2(T^{n+m}x)\ldots f_d(T^{n+(d-1)m}x)$$
converge $\mu$ a.e.
Deriving some results from the construction, for distal systems we answer positively the question if the multiple ergodic averages converge a.e. That is, we show that if $(X,\mathcal{X},\mu, T)$ is an ergodic distal system, and $f_1, \ldots, f_d \in L^{\infty}(\mu)$, then the multiple ergodic averages
$$\frac 1 N\sum_{n=0}^{N-1}f_1(T^nx)\ldots f_d(T^{dn}x)$$
converge $\mu$ a.e..
The two talks are mainly based on the following papers.
a. S. Shao and X.Ye, Regionally proximal relation of order d is an equivalence one for minimal systems and a combinatorial consequence,Adv. in Math.,231(2012), 1786-1817.
b. W. Huang, S. Shao and X. Ye,Nil Bohr-sets and almost automorphy of higher order, arXiv:1407.1179v1[math.DS], Memoirs of Amer. Math. Soc., to appear.
c. W. Huang, S. Shao and X Ye, Pointwise convergence of multiple ergodic averages and strictly ergodic models, arXiv:1406.5930v1[math.DS], submitted. |