The research of $({\rm{G}}, {\rm{w}})$-Chaos and G-Lipschitz shadowing property

Zhanjiang Ji; Zhanjiang Ji

doi:10.3934/math.2022566

AIMS Mathematics

2022, Volume 7, Issue 6: 10180-10194. doi: 10.3934/math.2022566

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The research of $({\rm{G}}, {\rm{w}})$-Chaos and G-Lipschitz shadowing property

Zhanjiang Ji ^{1,2,3
,
,}

1.
School of Data Science and Software Engineering, Wuzhou University, Wuzhou, Guangxi 543002, China
2.
Guangxi Colleges and Universities Key Laboratory of Image Processing and Intelligent Information System, Wuzhou University, Wuzhou, Guangxi 543002, China
3.
Guangxi Colleges and Universities Key Laboratory of Professional Software Technology, Wuzhou University, Wuzhou, Guangxi 543002, China

Received: 20 December 2021 Revised: 26 February 2022 Accepted: 06 March 2022 Published: 22 March 2022
MSC : 37B99

In this paper, we introduce the concepts of $ (G, w) - $ Chaos and $ G - $ Lipschitz shadowing property. We study the dynamical properties of $ (G, w) - $ Chaos in the inverse limit space under group action. In addition, we study the dynamical properties of $ G - $ Lipschitz shadowing property respectively under topological $ G - $ conjugate and iterative systems. The following conclusions are obtained. (1) Let $ ({X_f}, \bar G, {\text{ }}\bar d, \sigma) $ be the inverse limit space of $ (X, G, d, f) $ under group action. If the self-map $ f $ is $ (G, w) - $ chaotic, the shift map $ \sigma $ is $ (G, w) - $ chaotic; (2) Let $ (X, d) $ be a metric $ G - $ space and $ f $ be topologically $ G - $ conjugate to $ g $. Then the map $ f $ has $ G - $ Lipschitz shadowing property if and only if the map $ g $ has $ G - $ Lipschitz shadowing property. (3) Let $ (X, d) $ be a metric $ G - $ space and $ f $ be an equivariant Lipschitz map from $ X $ to $ X $. Then for any positive integer $ k \geqslant 2 $, the map $ f $ has the $ G - $ Lipschitz shadowing property if and only if the iterative map $ {f^k} $ has the $ G - $ Lipschitz shadowing property. These results enrich the theory of topological $ G - $ conjugate, iterative system and the inverse limit space under group action.
- inverse limit space,
- topological $ G - $ conjugate,
- iterative map,
- $ (G,w) - $ Chaos,
- $ G - $ Lipschitz shadowing property
Citation: Zhanjiang Ji. The research of $({\rm{G}}, {\rm{w}})$-Chaos and G-Lipschitz shadowing property[J]. AIMS Mathematics, 2022, 7(6): 10180-10194. doi: 10.3934/math.2022566

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Abstract

In this paper, we introduce the concepts of $ (G, w) - $ Chaos and $ G - $ Lipschitz shadowing property. We study the dynamical properties of $ (G, w) - $ Chaos in the inverse limit space under group action. In addition, we study the dynamical properties of $ G - $ Lipschitz shadowing property respectively under topological $ G - $ conjugate and iterative systems. The following conclusions are obtained. (1) Let $ ({X_f}, \bar G, {\text{ }}\bar d, \sigma) $ be the inverse limit space of $ (X, G, d, f) $ under group action. If the self-map $ f $ is $ (G, w) - $ chaotic, the shift map $ \sigma $ is $ (G, w) - $ chaotic; (2) Let $ (X, d) $ be a metric $ G - $ space and $ f $ be topologically $ G - $ conjugate to $ g $. Then the map $ f $ has $ G - $ Lipschitz shadowing property if and only if the map $ g $ has $ G - $ Lipschitz shadowing property. (3) Let $ (X, d) $ be a metric $ G - $ space and $ f $ be an equivariant Lipschitz map from $ X $ to $ X $. Then for any positive integer $ k \geqslant 2 $, the map $ f $ has the $ G - $ Lipschitz shadowing property if and only if the iterative map $ {f^k} $ has the $ G - $ Lipschitz shadowing property. These results enrich the theory of topological $ G - $ conjugate, iterative system and the inverse limit space under group action.

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