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( T i ( x ), T i ( y )) is greater than ε but for any positive δ this distance is less than δ for infinitely many i .