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<p>First we investigate relative <i>n</i>-regionally proximal tuples. Let <i>π</i>: (<i>X, G</i>) → (<i>Y, G</i>) be a Bronstein extension between minimal systems. It turns out that if (<i>x</i><sub>1</sub>,…, <i>x</i><sub><i>n</i></sub>) is a minimal point and (<i>x</i><sub><i>i</i></sub>, <i>x</i><sub><i>i</i>+1</sub>) is relative regionally proximal for 1 ≤ <i>i</i> ≤ <i>n</i> − 1, then (<i>x</i><sub>1</sub>,…, <i>x</i><sub><i>n</i></sub>) is relative <i>n</i>-regionally proximal. We consider the relative versions of sensitivity, including relative <i>n</i>-sensitivity and relative block ℱ<sub><i>t</i></sub>-<i>n</i>-sensitivity, where ℱ<sub><i>t</i></sub> is the family of thick sets. We show that <i>π</i> is relatively <i>n</i>-sensitive if and only if the relative <i>n</i>-regionally proximal relation contains a point whose coordinates are distinct, and the structure of <i>π</i> which is relatively <i>n</i>-sensitive but not relatively <i>n</i> + 1-sensitive is determined. We also characterize relatively block ℱ<sub><i>t</i></sub>-<i>n</i>-sensitive via relative regionally proximal tuples.</p>

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Relative Regionally Proximal Tuples and Sensitivity

  • 期刊论文

  • 发表日期：

    2025年8月15日

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  • Yini Yang

摘要

First we investigate relative n-regionally proximal tuples. Let π: (X, G) → (Y, G) be a Bronstein extension between minimal systems. It turns out that if (x1,…, xn) is a minimal point and (xi, xi+1) is relative regionally proximal for 1 ≤ i ≤ n − 1, then (x1,…, xn) is relative n-regionally proximal. We consider the relative versions of sensitivity, including relative n-sensitivity and relative block ℱt-n-sensitivity, where ℱt is the family of thick sets. We show that π is relatively n-sensitive if and only if the relative n-regionally proximal relation contains a point whose coordinates are distinct, and the structure of π which is relatively n-sensitive but not relatively n + 1-sensitive is determined. We also characterize relatively block ℱt-n-sensitive via relative regionally proximal tuples.