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 $1\leq i\leq k$, then $(x,x_1,\ldots,x_k)$ 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 $p_j^{(i)},1\leq i\leq k,$ $1\leq j\leq d$ such that $$ \bigcup_{i=1}^k\{ p_1^{(i)}\epsilon(1)+\ldots+p_d^{(i)} \epsilon(d):\epsilon(j)\in \{0,1\},1\leq j\leq d\}\backslash \{0\}\subset \mathrm{Ind}(U_0,U_1), $$ 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 $d$.