Non-convergence of some non-commuting double ergodic averages
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912118084730880 |
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| author | Austin, Tim |
| author_facet | Austin, Tim |
| contents | Let $S$ and $T$ be measure-preserving transformations of a probability space $(X,{\mathcal B},μ)$. Let $f$ be a bounded measurable functions, and consider the integrals of the corresponding `double' ergodic averages: \[\frac{1}{n}\sum_{i=0}^{n-1} \int f(S^ix)f(T^ix)\ dμ(x) \qquad (n\ge 1).\] We construct examples for which these integrals do not converge as $n\to\infty$. These include examples in which $S$ and $T$ are rigid, and hence have entropy zero, answering a question of Frantzikinakis and Host.
Our proof begins with a corresponding construction for orthogonal operators on a Hilbert space, and then obtains transformations of a Gaussian measure space from them. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_08630 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-convergence of some non-commuting double ergodic averages Austin, Tim Dynamical Systems Functional Analysis Primary: 37A30, Secondary: 47A35, 28A05, 37A44 Let $S$ and $T$ be measure-preserving transformations of a probability space $(X,{\mathcal B},μ)$. Let $f$ be a bounded measurable functions, and consider the integrals of the corresponding `double' ergodic averages: \[\frac{1}{n}\sum_{i=0}^{n-1} \int f(S^ix)f(T^ix)\ dμ(x) \qquad (n\ge 1).\] We construct examples for which these integrals do not converge as $n\to\infty$. These include examples in which $S$ and $T$ are rigid, and hence have entropy zero, answering a question of Frantzikinakis and Host. Our proof begins with a corresponding construction for orthogonal operators on a Hilbert space, and then obtains transformations of a Gaussian measure space from them. |
| title | Non-convergence of some non-commuting double ergodic averages |
| topic | Dynamical Systems Functional Analysis Primary: 37A30, Secondary: 47A35, 28A05, 37A44 |
| url | https://arxiv.org/abs/2407.08630 |