On proximality with Banach density one

Let (X,T)(X,T) be a topological dynamical system. A pair of points (x,y)∈X2(x,y)∈X2 is called Banach proximal if for any ε>0ε>0, the set {n∈Z+:d(Tnx,Tny)<ε}{n∈Z+:d(Tnx,Tny)<ε} has Banach density one. We study the structure of the Banach proximal relation. A useful tool is the notion of the support of a topological dynamical system. We show that a dynamical system is strongly proximal if and only if every pair in X2X2 is Banach proximal. A subset S of X is Banach scrambled if every two distinct points in S form a Banach proximal pair but not asymptotic. We construct a dynamical system with the whole space being a Banach scrambled set. Even though the Banach proximal relation of the full shift is of first category, it has a dense Mycielski invariant Banach scrambled set. We also show that for an interval map it is Li–Yorke chaotic if and only if it has a Cantor Banach scrambled set.