\(G\)-chain mixing and \(G\)-chain transitivity in metric \(G\)-space (Q2142344)

Summary: 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 \(k\geq2\), 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 \(G\left( U\right)=U, \overline{U}\neq X \), 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.