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).