Pointwise convergence of multiple ergodic averages and strictly ergodic models

Abstract

By building some suitable strictly ergodic models, we prove that for an ergodic system $(X,\mathcal{X},\mu ,T)$, $d\in \mathbb{N}$, $f_1,\dots ,f_d\in L^{\infty }(\mu )$, the averages $$\frac{1}{N^2}\sum _{(n,m)\in [0,N-1{]}^2}{f}_1(T^{n}x)f_2(T^{n+m}x)\dots f_d(T^{n+(d-1)m}x)$$ 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 $f_1,\dots ,f_d\in L^{\infty }(\mu )$, then multiple ergodic averages $$\frac{1}{N}\sum _{n=0}^{N-1}{f}_1(T^{n}x)\dots f_d(T^{dn}x)$$ converge $\mu$ a.e.

35 pages, revised version following referees' reports

References in corpus (2)

• On the pointwise convergence of multiple ergodic averages
• Pointwise recurrence for commuting measure preserving transformations

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• Topological correspondence of multiple ergodic averages of nilpotent group actions
• Weakly mixing, proximal topological models for ergodic systems and applications
• A cubic nonconventional ergodic average with multiplicative or Mangoldt weights
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• A good universal weight for multiple recurrence averages with commuting transformations in norm
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• Pointwise convergence of ergodic averages of bounded measurable functions for amenable groups
• Pointwise multiple averages for sublinear functions