On Transitive Points in a Generalized Shift Dynamical System

Abstract

Considering point transitive generalized shift dynamical system (XΓ,σφ) for discrete X with at least two elements and infinite Γ, we prove that X is countable and Γ has at most 2ℵ0 elements. Then, we find a transitive point of the dynamical system (NN×Z,στ) for τ:N×Z→N×Z with τ(n,m)=(n,m+1) and show that point transitive (XΓ,σφ), for infinite countable Γ, is a factor of (NN×Z,στ).