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a b s t r a c tIn 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 hs-periodic orbit is completely known as well as which order those points are visited, no matter how the hs-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.

The more self-crossing points an orbit has the more complex it is. We introduce the topological imprint to characterize crossing points and focus on the period-doubling cascade. The period-doubling cascade topological imprint determines the topological imprint for orbits in chaotic bands. In addition, there is a closer link between this concept and the braids studied by Lettelier et al (2000 J. Phys. A: Math. Gen. 33 1809–25).

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