# American Institute of Mathematical Sciences

February  2021, 41(2): 601-620. doi: 10.3934/dcds.2020278

## Maximal factors of order $d$ of dynamical cubespaces

 Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences and School of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China

Received  November 2019 Revised  June 2020 Published  August 2020

For a dynamical system $(X, T)$, $l\in\mathbb{N}$ and $x\in X$, let $\mathbf{Q}^{[l]}(X)$ and $\overline{\mathcal{F}^{[l]}}(x^{[l]})$ be the orbit closures of the diagonal point $x^{[l]}$ under the parallelepipeds group $\mathcal{G}^{[l]}$ and the face group $\mathcal{F}^{[l]}$ actions respectively. In this paper, it is shown that for a minimal system $(X, T)$ and every $l\in \mathbb{N}, x\in X$, the maximal factors of order $d$ of $(\mathbf{Q}^{[l]}(X), \mathcal{G}^{[l]})$ and $(\overline{\mathcal{F}^{[l]}}(x^{[l]}), \mathcal{F}^{[l]})$ are $(\mathbf{Q}^{[l]}(X_d), \mathcal{G}^{[l]})$ and $(\overline{\mathcal{F}^{[l]}}(\pi(x)^{[l]}), \mathcal{F}^{[l]})$ respectively, where $\pi:X\to X/\mathbf{RP}^{[d]}(X) = X_d, d\in \mathbb{N}\cup\{\infty\}$ is the factor map and $\mathbf{RP}^{[d]}(X)$ is the regionally proximal relation of order $d$.

Citation: Jiahao Qiu, Jianjie Zhao. Maximal factors of order $d$ of dynamical cubespaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 601-620. doi: 10.3934/dcds.2020278
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##### References:
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