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Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity
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 |
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 $.
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B. Host, B. Kra and A. Maass,
Nilsequences and a structure theorem for topological dynamical systems, Adv. in Math., 224 (2010), 103-129.
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J. Qiu and J. Zhao, Top-nilpotent enveloping semigroups and pro-nilsystems, arXiv: 1911.05435. |
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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. |
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J. de Vries, Elements of Topological Dynamics, Kluwer Academic Publishers Group, Dordrecht, 1993.
doi: 10.1007/978-94-015-8171-4. |
show all references
References:
| [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. |
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