Dynamical systems: recurrent and uniformly recurrent points

10.2. Example We consider the abstract dynamical system given in 9.18: the shift system over S where the phase space X is the product space {1, . . . , r}, for some r ≥ 2. For any R ⊆ S, there are x and U such that R(x, U) = R: simply pick x ∈ X be such that x(s) = 1 iff s ∈ R; so x might be considered as a characteristic function of R. Put U = {z ∈ X : z(1S) = 1}, a clopen subset of X. Then for s ∈ S, s ∈ R(x, U) holds iff (sx)(1S) = 1, which means that x(s) = 1, i.e. s ∈ R.