# 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  February 2021 Early access  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, 2021, 41 (2) : 601-620. doi: 10.3934/dcds.2020278
##### References:
 [1] J. Auslander, Minimal Flows and their Extensions, North-Holland Mathematics Studies, 153, North-Holland Publishing Co., Amsterdam, 1988.  Google Scholar [2] H. Becker and A. S. Kechris, The Descriptive Set Theory of Polish Groups Actions, London Math. Soc. Lecture Notes Ser., 232, Cambridge Univ. Press, Cambridge, 1996. doi: 10.1017/CBO9780511735264.  Google Scholar [3] P. Dong, S. Donoso, A. Maass, S. Shao and X. Ye, Infinite-step nilsystems, independence and complexity, Ergod. Th. and Dynam. Sys., 33 (2013), 118-143.  doi: 10.1017/S0143385711000861.  Google Scholar [4] B. Host and B. Kra, Personal communications., Google Scholar [5] B. Host and B. Kra, Nilpotent Structures in Ergodic Theory, Mathematical surveys and monographs 236, Providence, Rhode Island: American Mathematical Society, 2018.  Google Scholar [6] B. Host, B. Kra and A. Maass, Nilsequences and a structure theorem for topological dynamical systems, Adv. in Math., 224 (2010), 103-129.  doi: 10.1016/j.aim.2009.11.009.  Google Scholar [7] J. Qiu and J. Zhao, Top-nilpotent enveloping semigroups and pro-nilsystems, arXiv: 1911.05435. Google Scholar [8] S. Shao and X. Ye, Regionally proximal relation of order $d$ is an equivalence one for minimal systems and a combinatorial consequence, Adv. in Math., 231 (2012), 1786-1817.  doi: 10.1016/j.aim.2012.07.012.  Google Scholar [9] J. de Vries, Elements of Topological Dynamics, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-015-8171-4.  Google Scholar

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##### References:
 [1] J. Auslander, Minimal Flows and their Extensions, North-Holland Mathematics Studies, 153, North-Holland Publishing Co., Amsterdam, 1988.  Google Scholar [2] H. Becker and A. S. Kechris, The Descriptive Set Theory of Polish Groups Actions, London Math. Soc. Lecture Notes Ser., 232, Cambridge Univ. Press, Cambridge, 1996. doi: 10.1017/CBO9780511735264.  Google Scholar [3] P. Dong, S. Donoso, A. Maass, S. Shao and X. Ye, Infinite-step nilsystems, independence and complexity, Ergod. Th. and Dynam. Sys., 33 (2013), 118-143.  doi: 10.1017/S0143385711000861.  Google Scholar [4] B. Host and B. Kra, Personal communications., Google Scholar [5] B. Host and B. Kra, Nilpotent Structures in Ergodic Theory, Mathematical surveys and monographs 236, Providence, Rhode Island: American Mathematical Society, 2018.  Google Scholar [6] B. Host, B. Kra and A. Maass, Nilsequences and a structure theorem for topological dynamical systems, Adv. in Math., 224 (2010), 103-129.  doi: 10.1016/j.aim.2009.11.009.  Google Scholar [7] J. Qiu and J. Zhao, Top-nilpotent enveloping semigroups and pro-nilsystems, arXiv: 1911.05435. Google Scholar [8] S. Shao and X. Ye, Regionally proximal relation of order $d$ is an equivalence one for minimal systems and a combinatorial consequence, Adv. in Math., 231 (2012), 1786-1817.  doi: 10.1016/j.aim.2012.07.012.  Google Scholar [9] J. de Vries, Elements of Topological Dynamics, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-015-8171-4.  Google Scholar
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