On the convergence of multiple ergodic means
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866914186583343104 |
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| author | Karagulyan, Grigori A. Lacey, Michael T. Martirosyan, Vahan A. |
| author_facet | Karagulyan, Grigori A. Lacey, Michael T. Martirosyan, Vahan A. |
| contents | Given sequence of measure preserving transformations $\{U_k:\,k=1,2,\ldots, n\}$ on a measurable space $(X,μ)$. We prove a.e. convergence of the ergodic means
\begin{equation}
\frac{1}{s_1\cdots s_{n}}\sum_{j_1=0}^{s_1-1}\cdots\sum_{j_n=0}^{s_n-1}f\left(U_1^{j_1}\cdots U_n^{j_n} x \right)
\end{equation} as $\min_j s_j\to\infty $, for any function $f\in L\log^{d-1}(X)$, where $d\le n$ is the rank of the transformations. The result gives a generalization of a theorem by N. Dunford and A. Zygmund, claiming the convergence of the means in a narrower class of functions $L\log^{n-1}(X)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_00215 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the convergence of multiple ergodic means Karagulyan, Grigori A. Lacey, Michael T. Martirosyan, Vahan A. Classical Analysis and ODEs Dynamical Systems 37A30, 37A46, 42B25 Given sequence of measure preserving transformations $\{U_k:\,k=1,2,\ldots, n\}$ on a measurable space $(X,μ)$. We prove a.e. convergence of the ergodic means \begin{equation} \frac{1}{s_1\cdots s_{n}}\sum_{j_1=0}^{s_1-1}\cdots\sum_{j_n=0}^{s_n-1}f\left(U_1^{j_1}\cdots U_n^{j_n} x \right) \end{equation} as $\min_j s_j\to\infty $, for any function $f\in L\log^{d-1}(X)$, where $d\le n$ is the rank of the transformations. The result gives a generalization of a theorem by N. Dunford and A. Zygmund, claiming the convergence of the means in a narrower class of functions $L\log^{n-1}(X)$. |
| title | On the convergence of multiple ergodic means |
| topic | Classical Analysis and ODEs Dynamical Systems 37A30, 37A46, 42B25 |
| url | https://arxiv.org/abs/2208.00215 |