G -Chain Mixing and G -Chain Transitivity in Metric G-Space

Abstract

Firstly, we introduce the concept of $G$ -chain mixing, $G$ -mixing, and $G$ -chain transitivity in metric $G$ -space. Secondly, we study their dynamical properties and obtain the following results. (1) If the map $f$ has the $G$ -shadowing property, then the map $f$ is $G$ -chain mixed if and only if the map $f$ is $G$ -mixed. (2) The map $f$ is $G$ -chain transitive if and only if for any positive integer $k\ge 2$ , the map $f^k$ is $G$ -chain transitive. (3) If the map $f$ is $G$ -pointwise chain recurrent, then the map $f$ is $G$ -chain transitive. (4) If there exists a nonempty open set $U$ satisfying $G\left(U\right)=U$ , $\overline{U}\ne X$ , and $f\left(\overline{U}\right)\subset U$ , then we have that the map $f$ is not $G$ -chain transitive. These conclusions enrich the theory of $G$ -chain mixing, $G$ -mixing, and $G$ -chain transitivity in metric $G$ -space.