{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,12]],"date-time":"2026-02-12T10:29:13Z","timestamp":1770892153531,"version":"3.50.1"},"reference-count":34,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2019,9,25]],"date-time":"2019-09-25T00:00:00Z","timestamp":1569369600000},"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>A rigorous dynamical systems-based hierarchy is established for the definitions of entropy of Shannon (information), Kolmogorov\u2013Sinai (metric) and Adler, Konheim &amp; McAndrew (topological). In particular, metric entropy, with the imposition of some additional properties, is proven to be a special case of topological entropy and Shannon entropy is shown to be a particular form of metric entropy. This is the first of two papers aimed at establishing a dynamically grounded hierarchy comprising Clausius, Boltzmann, Gibbs, Shannon, metric and topological entropy in which each element is ideally a special case of its successor or some kind of limit thereof.<\/jats:p>","DOI":"10.3390\/e21100938","type":"journal-article","created":{"date-parts":[[2019,9,26]],"date-time":"2019-09-26T03:06:51Z","timestamp":1569467211000},"page":"938","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["A Dynamical Systems-Based Hierarchy for Shannon, Metric and Topological Entropy"],"prefix":"10.3390","volume":"21","author":[{"given":"Raymond","family":"Addabbo","sequence":"first","affiliation":[{"name":"Department of Arts and Sciences, Vaughn College of Aeronautics and Technology, Flushing, NY 11369, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3524-9538","authenticated-orcid":false,"given":"Denis","family":"Blackmore","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences and Center for Applied and Computational Mathematics, New Jersey Institute of Technology, Newark, NJ 07102-1982, USA"}]}],"member":"1968","published-online":{"date-parts":[[2019,9,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"493","DOI":"10.3390\/e10040493","article-title":"Entropy and uncertainty","volume":"10","author":"Robinson","year":"2008","journal-title":"Entropy"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"58","DOI":"10.3390\/entropy-e10020058","article-title":"Quantum and ecosystem entropies","volume":"10","author":"Kirwan","year":"2008","journal-title":"Entropy"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Denker, M., Grillenberger, C., and Sygmund, K. 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