American Institute of Mathematical Sciences

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

Maximal factors of order $ d $ of dynamical cubespaces

Jiahao Qiu and  Jianjie Zhao

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 $.

Keywords: Dynamical cubespaces, systems of order $ d $, nilsystems.
Mathematics Subject Classification: 37B05, 37A99.
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
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

show all references

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

[1]

Toshiko Ogiwara, Danielle Hilhorst, Hiroshi Matano. Convergence and structure theorems for order-preserving dynamical systems with mass conservation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3883-3907. doi: 10.3934/dcds.2020129
[2]

Mauricio Achigar. Extensions of expansive dynamical systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020399
[3]

The Editors. The 2019 Michael Brin Prize in Dynamical Systems. Journal of Modern Dynamics, 2020, 16: 349-350. doi: 10.3934/jmd.2020013
[4]

Nitha Niralda P C, Sunil Mathew. On properties of similarity boundary of attractors in product dynamical systems. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021004
[5]

Peter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein. Computing complete Lyapunov functions for discrete-time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 299-336. doi: 10.3934/dcdsb.2020331
[6]

Alessandro Fonda, Rodica Toader. A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo". Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021012
[7]

Stefan Siegmund, Petr Stehlík. Time scale-induced asynchronous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1011-1029. doi: 10.3934/dcdsb.2020151
[8]

Jérôme Lohéac, Chaouki N. E. Boultifat, Philippe Chevrel, Mohamed Yagoubi. Exact noise cancellation for 1d-acoustic propagation systems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020055
[9]

Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103
[10]

Xuefeng Zhang, Yingbo Zhang. Fault-tolerant control against actuator failures for uncertain singular fractional order systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 1-12. doi: 10.3934/naco.2020011
[11]

Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1627-1652. doi: 10.3934/dcdsb.2020176
[12]

Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002
[13]

Mikhail I. Belishev, Sergey A. Simonov. A canonical model of the one-dimensional dynamical Dirac system with boundary control. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021003
[14]

Oleg Yu. Imanuvilov, Jean Pierre Puel. On global controllability of 2-D Burgers equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 299-313. doi: 10.3934/dcds.2009.23.299
[15]

François Dubois. Third order equivalent equation of lattice Boltzmann scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 221-248. doi: 10.3934/dcds.2009.23.221
[16]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241
[17]

Yi An, Bo Li, Lei Wang, Chao Zhang, Xiaoli Zhou. Calibration of a 3D laser rangefinder and a camera based on optimization solution. Journal of Industrial & Management Optimization, 2021, 17 (1) : 427-445. doi: 10.3934/jimo.2019119
[18]

Balázs Kósa, Karol Mikula, Markjoe Olunna Uba, Antonia Weberling, Neophytos Christodoulou, Magdalena Zernicka-Goetz. 3D image segmentation supported by a point cloud. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 971-985. doi: 10.3934/dcdss.2020351
[19]

Xiaoli Lu, Pengzhan Huang, Yinnian He. Fully discrete finite element approximation of the 2D/3D unsteady incompressible magnetohydrodynamic-Voigt regularization flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 815-845. doi: 10.3934/dcdsb.2020143
[20]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (72)
  • HTML views (216)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences