American Institute of Mathematical Sciences

Advanced
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, 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]

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
[2]

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
[3]

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
[4]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074
[5]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020264
[6]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012
[7]

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
[8]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377
[9]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434
[10]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451
[11]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461
[12]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375
[13]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247
[14]

Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164
[15]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216
[16]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340
[17]

Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252
[18]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138
[19]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444
[20]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (44)
  • HTML views (146)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences