Kolmogorov–Sinai entropy from the ordinal viewpoint

In the case of ergodicity much of the structure of a one-dimensional time-discrete dynamical system is already determined by its ordinal structure. We generally discuss this phenomenon by considering the distribution of ordinal patterns, which describe the ups and downs in the orbits of a Borel measurable map on a Borel subset of the real line. On this base, we give a natural ordinal description of the Kolmogorov–Sinai entropy of one-dimensional dynamical systems and relate the Kolmogorov–Sinai entropy to the permutation entropy recently introduced by Bandt and Pompe.