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Pointwise convergence of multiple ergodic averages and strictly ergodic models

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Abstract

By building some suitable strictly ergodic models, we prove that for an ergodic system \((X, \mathcal{X}, \mu, T), d \in \mathbb{N}, f_1, \ldots, f_d \in L^\infty(\mu)\), the averages

$$\frac{1}{{{N^2}}}\sum\limits_{(n,m) \in {{[0,N - 1]}^2}} {{f_1}({T^n}x){f_2}({T^{n + m}}x) \cdots {f_d}({T^{n + (d - 1)m}}x)}$$

converge to a constant 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 \((X, \mathcal{X}, \mu, T)\) is an ergodic distal system, and f1, …, fd ∈ L∞(μ), then the multiple ergodic averages

$$\frac{1}{N}\sum\limits_{n = 0}^{N - 1} {{f_1}({T^n}x) \cdots {f_d}({T^{dn}}x)}$$

converge μ a.e.

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Authors and Affiliations

  1. Chinese Academy of Sciences and Department of Mathematics University of Science and Technology of China, Hefei, Anhui, 230026, P. R. China

    Wen Huang, Song Shao & Xiangdong Ye

Authors
  1. Wen Huang
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  2. Song Shao
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  3. Xiangdong Ye
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Corresponding author

Correspondence to Xiangdong Ye.

Additional information

Authors are supported by NNSF of China (11225105, 11371339, 11431012, 11571335) and by “The Fundamental Research Funds for the Central Universities”.

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Huang, W., Shao, S. & Ye, X. Pointwise convergence of multiple ergodic averages and strictly ergodic models. JAMA 139, 265–305 (2019). https://doi.org/10.1007/s11854-019-0061-3

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  • DOI: https://doi.org/10.1007/s11854-019-0061-3

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