Independence and almost automorphy of high order

Abstract

In this paper, it is shown that for a minimal system $(X,T)$ and $d,k\in \mathbb{N}$, if $(x,x_i)$ is regionally proximal of order $d$ for $1\le i\le k$, then $(x,x_1,\dots ,x_k)$ is $(k+1)$-regionally proximal of order $d$. Meanwhile, we introduce the notion of ${\mathrm{IN}}^{[d]}$-pair: for a dynamical system $(X,T)$ and $d\in \mathbb{N}$, a pair $(x_0,x_1)\in X\times X$ is called an ${\mathrm{IN}}^{[d]}$-pair if for any $k\in \mathbb{N}$ and any neighborhoods $U_0,U_1$ of $x_0$ and $x_1$ respectively, there exist integers $p_j^{(i)},1\le i\le k,$ $1\le j\le d$ such that $$\bigcup _{i=1}^k\{p_1^{(i)}\epsilon (1)+\dots +p_d^{(i)}\epsilon (d):\epsilon (j)\in \{0,1\},1\le j\le d\}\backslash \{0\}\subset \mathrm{Ind}(U_0,U_1),$$ where $\mathrm{Ind}(U_0,U_1)$ denotes the collection of all independence sets for $(U_0,U_1)$. It turns out that for a minimal system, if it dose not contain any nontrivial ${\mathrm{IN}}^{[d]}$-pair, then it is an almost one-to-one extension of its maximal factor of order $d$.

arXiv admin note: text overlap with arXiv:1911.05435. text overlap with arXiv:1105.3584, arXiv:1007.0189 by other authors