Abstract:

Let $(X, T)$ be a linear dynamical system, where $X$ is a separable Frechet space and $T:X \to X$ is a operator. first we prove the following assertions are equivalent:(1) $(X, T)$ is sensitive; (2) $(X, T)$ is multi-sensitive; (3) $(X, T)$ is multi-thickly-sensitive; (4) $(X, T)$ is thickly sensitive. Then we prove the following propositions:$(X, T)$ is syndetically sensitive if and only if $(X, T)$ is thickly syndetically sensitive. If $(X, T)$ is syndetic transitive, then $(X, T)$ is syndetically sensitive. If $(X, T)$ is frequently hypercyclic, then $(X, T)$ is thickly syndetically sensitive. If $(X, T)$ is syndetically transitive, then $(X, T)$ is transitively sensitive. Moreover, the iterated function system has the similar results.

Key words: Multi-sensitive, Thickly syndetically sensitive, Iterated function system