Topologically mildly mixing of higher orders along generalized polynomials
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866929202523013120 |
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| author | Cao, Yang Zhao, Jianjie |
| author_facet | Cao, Yang Zhao, Jianjie |
| contents | This paper is devoted to studying the multiple recurrent property of topologically mildly mixing systems along generalized polynomials. We show that if a minimal system is topologically mildly mixing, then it is mild mixing of higher orders along generalized polynomials. Precisely, suppose that $(X, T)$ is a topologically mildly mixing minimal system, $d\in \mathbb{N}$, $p_1, \dots, p_d$ are integer-valued generalized polynomials with $(p_1, \dots, p_d)$ non-degenerate. Then for all non-empty open subsets $U , V_1, \dots, V_d $ of $X$, $$\{n\in \Z: U\cap T^{-p_1(n) }V_1 \cap \dots \cap T^{-p_d(n) }V_d \neq \emptyset \}$$ is an IP$^*$-set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_03843 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Topologically mildly mixing of higher orders along generalized polynomials Cao, Yang Zhao, Jianjie Dynamical Systems This paper is devoted to studying the multiple recurrent property of topologically mildly mixing systems along generalized polynomials. We show that if a minimal system is topologically mildly mixing, then it is mild mixing of higher orders along generalized polynomials. Precisely, suppose that $(X, T)$ is a topologically mildly mixing minimal system, $d\in \mathbb{N}$, $p_1, \dots, p_d$ are integer-valued generalized polynomials with $(p_1, \dots, p_d)$ non-degenerate. Then for all non-empty open subsets $U , V_1, \dots, V_d $ of $X$, $$\{n\in \Z: U\cap T^{-p_1(n) }V_1 \cap \dots \cap T^{-p_d(n) }V_d \neq \emptyset \}$$ is an IP$^*$-set. |
| title | Topologically mildly mixing of higher orders along generalized polynomials |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2401.03843 |