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

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 $(X_f,\overline{G},\overline{d},\sigma )$ 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 $k⩾2$, 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.