Bulletin of the
Korean Mathematical Society BKMS

Let $\{T_t\}_{t\in \Delta}$ be the translation semigroup with a sector $\Delta\subset \mathbb{C}$ as index set. The recurrent hypercyclicity criterion (RHCC) for the $C_0$-semigroup $\{T_t\}_{t\in \Delta}$ is established, and then the equivalent conditions ensuring $\{T_t\}_{t\in \Delta}$ satisfying the RHCC on weighted spaces of $p$-integrable and of continuous functions are presented. Especially, every chaotic semigroup $\{T_t\}_{t\in \Delta}$ satisfies the RHCC.

Keywords: Hypercyclicity, translation semigroup, recurrent hypercyclicity criterion

MSC numbers: Primary 37B20, 47A16, 47D60

Supported by: The first author was supported in part by the National Natural Science Foundation of China Grant \#11701422. The second author was supported in part by the National Natural Science Foundation of China Grant \#12271396. The third author was supported in part by the National Natural Science Foundation of China Grant \#12171353..