Let X and P be compact topological spaces and let \(\pi_ X\) and \(\pi_ P\) be the projections of the product space \(X\times P\) onto X and P, respectively. Let \(F:X\times P\to X\times P\) be a continuous mapping and let there exist a mapping \(f:P\to P\) such that for all \((x,p)\in X\times P:\) \(\pi_ PF(x,p)=f(p).\) Let \(\hat p\in P\) be a fixed point of f. Define a mapping \(\hat g:X\to X\) by letting \(\hat g(x)=\pi_ XF(x,\hat p).\) Thus, one has three discrete semi-dynamical systems induced by F (on \(X\times P)\), f (on P) and \(\hat g \)(on X), respectively. The paper is an exhaustive study on relations between convergence properties of these three systems.