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.
Highlights
Determining the Kolmogorov-Sinai entropy (K-S entropy) of a dynamical system is a key part in the analysis of a system’s complexity
We show that the equality of K-S and permutation entropy still holds true for one-dimensional systems if one omits the condition of continuity assumed in [2] and allows a countable partition of the domain of definition into intervals such that the one-dimensional map is monotone on each of those intervals
For the definition of the permutation entropy, we investigate for which s, t ∈ N0 the inequality T s(ω) ≤ T t(ω) holds true
Summary
Determining the Kolmogorov-Sinai entropy (K-S entropy) of a dynamical system is a key part in the analysis of a system’s complexity. In their seminal paper [2], Bandt, Pompe and G Keller showed that those entropies are equal for one-dimensional interval maps if there exists a finite partition of the domain of definition into intervals such that the considered map is monotone and continuous on each of those intervals [2]. This result leads to the generally mathematically interesting question of the relationship of K-S entropy and permutation entropy. Our paper should be understood as a step to answering this question
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