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. DongS. DonosoA. MaassS. 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. HostB. 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. DongS. DonosoA. MaassS. 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. HostB. 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]

Sebastián Donoso. Enveloping semigroups of systems of order d. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2729-2740. doi: 10.3934/dcds.2014.34.2729

[2]

Ahmed Y. Abdallah. Exponential attractors for second order lattice dynamical systems. Communications on Pure & Applied Analysis, 2009, 8 (3) : 803-813. doi: 10.3934/cpaa.2009.8.803

[3]

Xiangjin Xu. Multiple solutions of super-quadratic second order dynamical systems. Conference Publications, 2003, 2003 (Special) : 926-934. doi: 10.3934/proc.2003.2003.926

[4]

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

[5]

Xiaojun Chang, Yong Li. Rotating periodic solutions of second order dissipative dynamical systems. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 643-652. doi: 10.3934/dcds.2016.36.643

[6]

Bin Wang, Arieh Iserles. Dirichlet series for dynamical systems of first-order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 281-298. doi: 10.3934/dcdsb.2014.19.281

[7]

H. W. Broer, Renato Vitolo. Dynamical systems modeling of low-frequency variability in low-order atmospheric models. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 401-419. doi: 10.3934/dcdsb.2008.10.401

[8]

Martin Redmann, Melina A. Freitag. Balanced model order reduction for linear random dynamical systems driven by Lévy noise. Journal of Computational Dynamics, 2018, 5 (1&2) : 33-59. doi: 10.3934/jcd.2018002

[9]

Adolfo Damiano Cafaro, Simone Fiori. Optimization of a control law to synchronize first-order dynamical systems on Riemannian manifolds by a transverse component. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021213

[10]

El Houcein El Abdalaoui, Sylvain Bonnot, Ali Messaoudi, Olivier Sester. On the Fibonacci complex dynamical systems. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2449-2471. doi: 10.3934/dcds.2016.36.2449

[11]

Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete & Continuous Dynamical Systems, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355

[12]

Mauricio Achigar. Extensions of expansive dynamical systems. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3093-3108. doi: 10.3934/dcds.2020399

[13]

Fritz Colonius, Marco Spadini. Fundamental semigroups for dynamical systems. Discrete & Continuous Dynamical Systems, 2006, 14 (3) : 447-463. doi: 10.3934/dcds.2006.14.447

[14]

John Erik Fornæss. Sustainable dynamical systems. Discrete & Continuous Dynamical Systems, 2003, 9 (6) : 1361-1386. doi: 10.3934/dcds.2003.9.1361

[15]

Vieri Benci, C. Bonanno, Stefano Galatolo, G. Menconi, M. Virgilio. Dynamical systems and computable information. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 935-960. doi: 10.3934/dcdsb.2004.4.935

[16]

Mădălina Roxana Buneci. Morphisms of discrete dynamical systems. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 91-107. doi: 10.3934/dcds.2011.29.91

[17]

Josiney A. Souza, Tiago A. Pacifico, Hélio V. M. Tozatti. A note on parallelizable dynamical systems. Electronic Research Announcements, 2017, 24: 64-67. doi: 10.3934/era.2017.24.007

[18]

Philippe Marie, Jérôme Rousseau. Recurrence for random dynamical systems. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 1-16. doi: 10.3934/dcds.2011.30.1

[19]

Tobias Wichtrey. Harmonic limits of dynamical systems. Conference Publications, 2011, 2011 (Special) : 1432-1439. doi: 10.3934/proc.2011.2011.1432

[20]

Yejuan Wang, Chengkui Zhong, Shengfan Zhou. Pullback attractors of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems, 2006, 16 (3) : 587-614. doi: 10.3934/dcds.2006.16.587

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (168)
  • HTML views (244)
  • Cited by (0)

Other articles
by authors

[Back to Top]