Composition law of cardinal ordering permutations

In this paper the theorems that determine composition laws both cardinal ordering permutations and their inverses are proven. So, the relative position of points in an h s -periodic orbit is completely known as well as which order those points are visited, no matter how the h s -periodic orbit emerges, be it through a period doubling cascade ( s = 2 n ) of as the h -periodic orbit or a primary window (like saddle-node bifurcation cascade with h = 2 n ) or a secondary window (the birth of an s -periodic window inside the h -periodic one). Certainly, period doubling cascade orbits are particular cases with h = 2 and s = 2 n . Both composition laws are also shown in algorithmic way for easy use.