PSU Mark
Eberly College of Science Mathematics Department

Meeting Details

For more information about this meeting, contact Anatole Katok, Federico Rodriguez Hertz, Boris Kalinin.

Title:Pointwise convergence of multiple ergodic averages and strictly ergodic models
Seminar:Working Seminar: Dynamics and its Working Tools
Speaker:Xiangdong Ye, University of Science and Technology of China
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.

Room Reservation Information

Room Number:MB216
Date:11 / 04 / 2014
Time:03:30pm - 06:00pm