Dynamics of the translation semigroup on directed metric trees

The dynamics of the left translation semigroup $\{T_t\}_{t \geq 0}$ on weighted $L^p$ spaces over a directed metric tree $L(G)$ is investigated. Necessary and sufficient conditions on the weight family $\rho$ for the strong continuity of the semigroup are provided. Furthermore, hypercyclicity and weak mixing properties are characterized in terms of the asymptotic decay of $\rho$ along the tree structure. These results generalize classical $L^p$ translation semigroup dynamics to a graph setting.