Topological correspondence of multiple ergodic averages of nilpotent group actions

2016/04/25 by Huang, Wen, Shao, Song, Ye, Xiangdong · 1 citation
#37A25 #37B05 #37B20 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1604.07113

Abstract

Let (X,Γ) be a topological system, where Γ is a nilpotent group generated by T₁,…, T_d such that for each T∈ Γ, T≠ e_Γ, (X,T) is weakly mixing and minimal. For d,k∈ ℕ, let p_i,j(n), 1≤ i≤ k, 1≤ j≤ d be polynomials with rational coefficients taking integer values on the integers and p_i,j(0)=0. We show that if the expressions g_i(n)=T₁^p_i,1(n)⋯ T_d^p_i,d(n) depends nontrivially on n for i=1,2,⋯,k, and for all i≠ j∈ \1,2,…,k\ the expressions g_i(n)g_j(n)^-1 depend nontrivially on n, then there is a residual set X₀ of X such that for all x∈ X₀ \(g₁(n)x, g₂(n)x,…, g_k(n)x)∈ X^k:n∈ ℤ\ is dense in X^k.

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• Weighted Birkhoff averages: Deterministic and probabilistic perspectives

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