pointwise almost periodicity in a generalized shift dynamical system

In the following text we prove that in generalized shift dynamical system (x^gamma,sigma_varphi) for discrete x with at least two elements, arbitrary nonempty gamma and selfmap varphi:gammatogamma , the following statements are equivalent: 1) (x^gamma,sigma_varphi) is pointwise recurrent; 2) (x^gamma,sigma_varphi) is pointwise almost periodic; 3) (x^gamma,sigma_varphi) is pointwise regularly almost periodic; 4) (x^gamma,sigma_varphi) is compactly almost periodic; 5) {rm per}(varphi)=gamma ( varphi:gammatogamma is pointwise periodic).