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Topological correspondence of multiple ergodic averages of nilpotent group actions

  • Wen Huang
  • Song Shao
  • Xiangdong YeEmail author
Article

Abstract

Let (X,Γ) be a topological system, where Γ is a nilpotent group generated by T1,...,Td such that for each T ∈ Γ, TeΓ, (X,T) is weakly mixing and minimal. For d,k ∈ ℕ, let pi,j(n),1 ≤ ik,1 ≤ jd be polynomials with rational coefficients taking integer values on the integers and pi,j(0) = 0. We show that if the expressions \(g_i(n)=T_1^{{p}_{i,1}(n)}\cdots{T_d^{{p}_{i,d}(n)}}\) depend nontrivially on n for i = 1,2,...,k, and for all ij ∈ {1,2,...,k} the expressions gi(n)gj(n)-1 depend nontrivially on n, then there is a residual set X0 of X such that for all xX0
$$\{(g_1(n)x, g_2(n)x, ...., g_k(n)x)\in{X^k}:n\in\mathbb{Z}\}$$
is dense in Xk.

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© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity Of Science and Technology of China, Chinese Academy of SciencesHefei, AnhuiP. R. China

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