Frequently hypercyclic and Devaney chaotic $C_0$-semigroups indexed by complex sectors

In this paper, the definitions of frequent hypercyclicity and Devaney chaos for a $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathrm{\Delta }}$ indexed by a complex sector $\mathrm{\Delta }$ are revisited. A general sufficient criterion and necessary conditions for a $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathrm{\Delta }}$ to be frequently hypercyclic are established. By adapting the concept of periodic point with a syndetic set of periods, we present clean and effective characterizations of the Devaney chaotic translation semigroup ${\left\{T_t\right\}}_{t\in \mathrm{\Delta }}$ on $L_{\rho }^p(\mathrm{\Delta },\mathbb{K}),1\le p<\mathrm{\infty }$, which generalize those of the classical translation semigroup ${\left\{T_t\right\}}_{t\ge 0}$. Moreover, it is shown that Devaney chaos implies frequent hypercyclicity, topological mixing, and distributional chaos for the translation semigroups ${\left\{T_t\right\}}_{t\in \mathrm{\Delta }}$ under our revised definitions, and that topological mixing or distributional chaos does not imply Devaney chaos.