Furstenberg Families and Sensitivity

We introduce and study some concepts of sensitivity via Furstenberg families. A dynamical system (X, f) is F-sensitive if there exists a positive e such that for every x is an element of X and every open neighborhood U of x there exists y is an element of U such that the pair (x, y) is not F-epsilon-asymptotic; that is, the time set {n : d (f(n)(x ), f(n) (y)) > epsilon} belongs to F, where F is a Furstenberg family. A dynamical system (X, f) is (F-1, F-2)-sensitive if there is a positive epsilon such that every x is an element of X is a limit of points y. X such that the pair (x, y) is F-1-proximal but not F-2-epsilon-asymptotic; that is, the time set {n : d(f(n)(x), f(n) (y)) < delta} belongs to F-1 for any positive d but the time set {n : d(f(n)(x), f(n) (y)) < epsilon} belongs to F-2, where F-1 and F-2 are Furstenberg families.