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arXiv · 2401.03843

Topologically mildly mixing of higher orders along generalized polynomials

Yang Cao·Jianjie Zhao
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Abstract

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.

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Yang Cao, Jianjie Zhao. 2024-01-08. Topologically mildly mixing of higher orders along generalized polynomials. https://arxiv.org/abs/2401.03843

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