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

<abstract> <p>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.</p> </abstract>