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 ψψ on a manifold M   (i.e., a continuous semi-flow ψψ on a manifold M   satisfies that limt→+∞|Ct(φ,ϕ,ψ)|=0limt→+∞|Ct(φ,ϕ,ψ)|=0 for any two continuous functions φ:M→Rφ:M→R and ϕ:M→Rϕ:M→R) is sensitive and topologically transitive. Furthermore, we show that a chaotic semi-flow ψψ 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 ψ:R+×M→Mψ:R+×M→M is a C1C1 semi-flow such that for any r>0,ψrr>0,ψr satisfies the chaotic definition of Devaney, and if for any r>0r>0, ‖Dψr(x)·v‖‖v‖≠1,for any x∈Mx∈M and any v∈TxMv∈TxM, then ψψ 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).