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 ε such that for every x∈X and every open neighborhood U of x there exists y∈U such that the pair (x,y) is not ℱ-ε-asymptotic; that is, the time set {n:d(fn(x),fn(y))>ε} belongs to ℱ, where ℱ is a Furstenberg family. A dynamical system (X,f) is (ℱ1, ℱ2)-sensitive if there is a positive ε such that every x∈X is a limit of points y∈X such that the pair (x,y) is ℱ1-proximal but not ℱ2-ε-asymptotic; that is, the time set {n:d(fn(x),fn(y))ε} belongs to ℱ2, where ℱ1 and ℱ2 are Furstenberg families.