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A pointwise cubic average for two commuting transformations
Dynamical Systems · Submitted 17 Oct 2014
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Using a notion of dynamical cubes introduced by the authors, the Huang-Shao-Ye technique and the Host machinery of magic systems, we prove that for a system $(X,\mu,S,T)$ with commuting transformations $S$ and $T$, the average \[\frac{1}{N^2} \sum_{i,j=0}^{N-1} f_0(S^i x)f_1(T^j x)f_2(S^i T^j x)\] converges a.e. as $N$ goes to infinity for any $f_1,f_2,f_3\in L^{\infty}(\mu)$.
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The abstract, sentence by sentence
Huang, Shao and Ye recently studied pointwise multiple averages by using suitable topological models. Using a notion of dynamical cubes introduced by the authors, the Huang-Shao-Ye technique and the Host machinery of magic systems, we prove that for a system $(X,\mu,S,T)$ with commuting transformations $S$ and $T$, the average \[\frac{1}{N^2} \sum_{i,j=0}^{N-1} f_0(S^i x)f_1(T^j x)f_2(S^i T^j x)\] converges a.e. as $N$ goes to infinity for any $f_1,f_2,f_3\in L^{\infty}(\mu)$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Huang, Shao and Ye recently studied pointwise multiple averages by using suitable topological models. | background 1.00 | 0.04 | 0.02 | asserted 0.99 |
| s1 | Using a notion of dynamical cubes introduced by the authors, the Huang-Shao-Ye technique and the Host machinery of magic systems, we prove that for a system $(X,\mu,S,T)$ with commuting transformations $S$ and $T$, the average \[\frac{1}{N^2} \sum_{i,j=0}^{N-1} f_0(S^i x)f_1(T^j x)f_2(S^i T^j x)\] converges a.e. as $N$ goes to infinity for any $f_1,f_2,f_3\in L^{\infty}(\mu)$. | result 0.79 | 0.95 | 0.96 | asserted 1.00 |
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The paper
Original abstract
Huang, Shao and Ye recently studied pointwise multiple averages by using suitable topological models. Using a notion of dynamical cubes introduced by the authors, the Huang-Shao-Ye technique and the Host machinery of magic systems, we prove that for a system $(X,\mu,S,T)$ with commuting transformations $S$ and $T$, the average \[\frac{1}{N^2} \sum_{i,j=0}^{N-1} f_0(S^i x)f_1(T^j x)f_2(S^i T^j x)\] converges a.e. as $N$ goes to infinity for any $f_1,f_2,f_3\in L^{\infty}(\mu)$.
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1 arXiv paper cite this in 1 sentence
Papers citing it
Cites it as background 1 paper
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Many generalization, in different contexts have been found for the Jewett-Krieger Theorem [ref] and very recently several applications have been given for the pointwise convergence of different ergodic averages [ref] and to build interesting examples in topological dynamics [ref] .
Uniformly rigid models for rigid actions · Measurable and topological systems · 0.92
What it cites · 4 sentences without a confident label
Uses the method or tool of 1 paper
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Then we use the method of Huang, Shao and Ye [ref] to find a desired model for this extension system in Section 4.
Strictly ergodic models and the convergence of non-conventional pointwise ergodic averages · Proof Strategy and organization · 1.00
Cites as background 1 paper
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Recently Huang, Shao and Ye [ref] proved the pointwise convergence of multiple averages for a single transformation on a distal system.
Pointwise convergence of multiple ergodic averages and strictly ergodic models · Pointwise convergence for cube averages · 0.83
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