Kolmogorov-Sinai entropy from the ordinal viewpoint

TL;DR

This paper explores how the ordinal structure of one-dimensional dynamical systems can determine their Kolmogorov-Sinai entropy, linking it to permutation entropy through the distribution of ordinal patterns.

Contribution

It provides a natural ordinal description of Kolmogorov-Sinai entropy and relates it to permutation entropy for a broad class of systems.

Findings

Ordinal patterns capture key dynamical information.

Kolmogorov-Sinai entropy can be described via ordinal structures.

Relation established between Kolmogorov-Sinai and permutation entropy.

Abstract

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 up and down in the orbits of a Borel measurable map on a subset of the real numbers. In particular, we give a natural ordinal description of Kolmogorov-Sinai entropy of a large class of one-dimensional dynamical systems and relate Kolmogorov-Sinai entropy to the permutation entropy recently introduced by Bandt and Pompe.