The realization and classification of topologically transitive group actions on $1$-manifolds

In this report, we first recall the Poincares classification theorem for minimal orientation-preserving homeomorphisms on the circle and the Ghys classification theorem for minimal orientation-preserving group actions on the circle. Then we introduce a classification theorem for a specified class of topologically transitive orientation-preserving group actions on the circle by $mathbb Z^d$. Also, some groups that admit/admit no topologically transitive actions on the line are determined.