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Article Contents

# Maximal factors of order $d$ of dynamical cubespaces

• 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 $d$.

Mathematics Subject Classification: 37B05, 37A99.

 Citation:

•  [1] J. Auslander, Minimal Flows and their Extensions, North-Holland Mathematics Studies, 153, North-Holland Publishing Co., Amsterdam, 1988. [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. [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. [4] B. Host and B. Kra, Personal communications., [5] B. Host and B. Kra, Nilpotent Structures in Ergodic Theory, Mathematical surveys and monographs 236, Providence, Rhode Island: American Mathematical Society, 2018. [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. [7] J. Qiu and J. Zhao, Top-nilpotent enveloping semigroups and pro-nilsystems, arXiv: 1911.05435. [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. [9] J. de Vries, Elements of Topological Dynamics, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-015-8171-4.