Devaney chaos revisited

Let X be a metric space, and let f : X → X be a continuous transformation. In this note, a concept indecomposability of f is introduced. We show that transitivity implies indecomposability and that Devaney chaos is equivalent to indecomposability together with dense periodicity. Moreover, we point out that the indecomposability and the dense periodicity are independent of each other even for interval maps (i.e., neither implies the other).