A note on decay of correlation implies chaos in the sense of Devaney

In this paper, we prove that a mixing (in the sense of statistics), continuous semi-flow psi on a manifold M (i.e., a continuous semi-flow 0 on a manifold M satisfies that lim(t ->+infinity)vertical bar C-t(phi, phi, psi)vertical bar= 0 for any two continuous functions phi : M -> R and phi : M -> R) is sensitive and topologically transitive. Furthermore, we show that a chaotic semi-flow 0 on a manifold M in the sense of Devaney with some assumptions is an expanding (in the sense of differentiable dynamical system) semi-flow, that is, if psi : R+ x M -> M is a C-1 semi-flow such that for any r > 0, satisfies the chaotic definition of Devaney, and if for any r > 0, parallel to D psi(r)(x) . nu parallel to/parallel to nu parallel to not equal 1, for any x is an element of M and any nu is an element of TxM, then psi is expanding (in the sense of differentiable dynamical system). Also, we prove that a continuous selfmap of a compact metric space satisfies the Devaney's definition of chaos if and only if the same holds for the suspended semi-flow induced by it, and that if a continuous selfmap of a compact metric space is mixing (in the sense of statistics) if and only if so is the suspended semi-flow induced by it. The above results improve and extend the corresponding results in Xu et al. (2004). (C) 2015 Elsevier Inc. All rights reserved.