Several transitive properties and Devaney's chaos

The relation among transitivity, indecomposability and .-transitivity is discussed. It is shown that for a non-wandering system (each point is non-wandering), indecomposability is equivalent to transitivity, and for the dynamical systems without isolated points, .-transitivity and transitivity are equivalent. Besides, a new transitive level as weak transitivity is introduced and some equivalent conditions of Devaney.s chaos are given by weak transitivity. Moreover, it is proved that both d-shadowing property and d-shadowing property imply weak transitivity.