{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,20]],"date-time":"2026-03-20T23:57:02Z","timestamp":1774051022027,"version":"3.50.1"},"reference-count":16,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,1,2]],"date-time":"2020-01-02T00:00:00Z","timestamp":1577923200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Different authors have shown strong relationships between ordinal pattern based entropies and the Kolmogorov\u2013Sinai entropy, including equality of the latter one and the permutation entropy, the whole picture is however far from being complete. This paper is updating the picture by summarizing some results and discussing some mainly combinatorial aspects behind the dependence of Kolmogorov\u2013Sinai entropy from ordinal pattern distributions on a theoretical level. The paper is more than a review paper. A new statement concerning the conditional permutation entropy will be given as well as a new proof for the fact that the permutation entropy is an upper bound for the Kolmogorov\u2013Sinai entropy. As a main result, general conditions for the permutation entropy being a lower bound for the Kolmogorov\u2013Sinai entropy will be stated. Additionally, a previously introduced method to analyze the relationship between permutation and Kolmogorov\u2013Sinai entropies as well as its limitations will be investigated.<\/jats:p>","DOI":"10.3390\/e22010063","type":"journal-article","created":{"date-parts":[[2020,1,3]],"date-time":"2020-01-03T04:43:03Z","timestamp":1578026583000},"page":"63","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Ordinal Pattern Based Entropies and the Kolmogorov\u2013Sinai Entropy: An Update"],"prefix":"10.3390","volume":"22","author":[{"given":"Tim","family":"Gutjahr","sequence":"first","affiliation":[{"name":"Institute of Mathematics, University of L\u00fcbeck, D-23562 L\u00fcbeck, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6321-0465","authenticated-orcid":false,"given":"Karsten","family":"Keller","sequence":"additional","affiliation":[{"name":"Institute of Mathematics, University of L\u00fcbeck, D-23562 L\u00fcbeck, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,1,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1595","DOI":"10.1088\/0951-7715\/15\/5\/312","article-title":"Entropy of interval maps via permutations","volume":"15","author":"Bandt","year":"2002","journal-title":"Nonlinearity"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"789","DOI":"10.1016\/j.physd.2012.01.004","article-title":"The equality of Kolmogorov\u2013Sinai entropy and metric permutation entropy generalized","volume":"241","year":"2012","journal-title":"Physica D"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1793","DOI":"10.3934\/dcds.2014.34.1793","article-title":"Kolmogorov\u2013Sinai entropy via separation properties of order-generated \u03c3-algebras","volume":"34","author":"Antoniouk","year":"2014","journal-title":"Discret. 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(2010). Ergodic Theory: With a View Towards Number Theory, Springer. Graduate Texts in Mathematics.","DOI":"10.1007\/978-0-85729-021-2"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1477","DOI":"10.1016\/j.physd.2012.05.010","article-title":"On the relation of KS entropy and permutation entropy","volume":"241","author":"Keller","year":"2012","journal-title":"Physica D"},{"key":"ref_13","unstructured":"Walters, P. (2000). An Introduction to Ergodic Theory, Springer. Graduate Texts in Mathematics."},{"key":"ref_14","first-page":"55","article-title":"On the fundamental ideas of measure theory","volume":"71","author":"Rokhlin","year":"1952","journal-title":"Am. Math. Soc. Transl."},{"key":"ref_15","first-page":"201","article-title":"Rokhlin\u2019s Lemma for Non-invertible Maps","volume":"2","author":"Heinemann","year":"2001","journal-title":"Dyn. Syst. Appl."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Halmos, P.R. (1950). Extension of Measures. 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