On proximality with Banach density one

Let (X,T)$(X,T)$ be a topological dynamical system. A pair of points (x,y)∈X²$(x,y)\in X^2$ is called Banach proximal if for any ε>0$\epsilon >0$, the set {n∈_Z+:d(Tⁿx,Tⁿy)<ε}$\{n\in {\mathbb{Z}}_{+}:d(T^{n}x,T^{n}y)<\epsilon \}$ 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 X²$X^2$ 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.