Equality of Kolmogorov-Sinai and permutation entropy for one-dimensional maps consisting of countably many monotone parts

TL;DR

This paper proves that for certain one-dimensional maps with countably many monotone segments, the Kolmogorov-Sinai entropy equals the permutation entropy, extending previous results that required finitely many segments and continuity.

Contribution

It generalizes the equality of Kolmogorov-Sinai and permutation entropy to maps with countably many monotone parts, relaxing previous finiteness and continuity assumptions.

Findings

Kolmogorov-Sinai and permutation entropy are equal under the given conditions.

The result extends previous finite-interval cases to countably infinite partitions.

Provides a broader understanding of entropy equivalence in dynamical systems.

Abstract

In this paper, we show that, under some technical assumptions, the Kolmogorov-Sinai entropy and the permutation entropy are equal for one-dimensional maps if there exists a countable partition of the domain of definition into intervals such that the considered map is monotone on each of those intervals. This is a generalization of a result by Bandt, Pompe and G. Keller, who showed that the above holds true under the additional assumptions that the number of intervals on which the map is monotone is finite and that the map is continuous on each of those intervals.