Pointwise convergence in nilmanifolds along smooth functions of polynomial growth
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866916487261847552 |
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| author | Tsinas, Konstantinos |
| author_facet | Tsinas, Konstantinos |
| contents | We study the equidistribution of orbits of the form $b_1^{a_1(n)}... b_k^{a_k(n)}Γ$ in a nilmanifold $X$, where the sequences $a_i(n)$ arise from smooth functions of polynomial growth belonging to a Hardy field. We show that under certain assumptions on the growth rates of the functions $a_1,...,a_k$, these orbits are uniformly distributed on some subnilmanifold of the space $X$. As an application of these results and in combination with the Host-Kra structure theorem for measure preserving systems, as well as some recent seminorm estimates of the author for ergodic averages concerning Hardy field functions, we deduce a norm convergence result for multiple ergodic averages. Our method mainly relies on an equidistribution result of Green-Tao on finite polynomial orbits of a nilmanifold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_11609 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Pointwise convergence in nilmanifolds along smooth functions of polynomial growth Tsinas, Konstantinos Dynamical Systems 22F30(Primary), 37A17 (Secondary) We study the equidistribution of orbits of the form $b_1^{a_1(n)}... b_k^{a_k(n)}Γ$ in a nilmanifold $X$, where the sequences $a_i(n)$ arise from smooth functions of polynomial growth belonging to a Hardy field. We show that under certain assumptions on the growth rates of the functions $a_1,...,a_k$, these orbits are uniformly distributed on some subnilmanifold of the space $X$. As an application of these results and in combination with the Host-Kra structure theorem for measure preserving systems, as well as some recent seminorm estimates of the author for ergodic averages concerning Hardy field functions, we deduce a norm convergence result for multiple ergodic averages. Our method mainly relies on an equidistribution result of Green-Tao on finite polynomial orbits of a nilmanifold. |
| title | Pointwise convergence in nilmanifolds along smooth functions of polynomial growth |
| topic | Dynamical Systems 22F30(Primary), 37A17 (Secondary) |
| url | https://arxiv.org/abs/2203.11609 |