Li–Yorke sensitivity

We introduce and study a concept which links the Li–Yorke versions of chaos with the notion of sensitivity to initial conditions. We say that a dynamical system (X,T) is Li–Yorke sensitive if there exists a positive ε such that every x∊X is a limit of points y∊X such that the pair (x,y) is proximal but not ε-asymptotic, i.e. for infinitely many positive integers i the distance ρ(Tⁱ(x),Tⁱ(y)) is greater than ε but for any positive δ this distance is less than δ for infinitely many i.