Pointwise convergence of multiple ergodic averages and strictly ergodic models

2014/06/23 by Huang, Wen, Shao, Song, Ye, Xiangdong · 2 citations
#37A05 #37B05 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1406.5930

Abstract

By building some suitable strictly ergodic models, we prove that for an ergodic system (X,X,μ, T), d∈\mathbb N, f₁, …, f_d ∈ L^∞(μ), the averages (1)/(N²) ∑_{(n,m)∈ [0,N-1]²} f₁(Tⁿx)f₂(T^n+mx)… f_d(T^n+(d-1)mx) converge μ 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,X,μ, T) is an ergodic distal system, and f₁, …, f_d ∈ L^∞(μ), then multiple ergodic averages \frac 1 N∑_n=0^N-1f₁(Tⁿx)… f_d(T^dnx) converge μ a.e.

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• More stability and convergence results for higher-order Wiener-Wintner systems
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