A Dynamical Systems-Based Hierarchy for Shannon, Metric and Topological Entropy
A rigorous dynamical systems-based hierarchy is established for the definitions of entropy of Shannon (information), Kolmogorov−Sinai (metric) and Adler, Konheim & McAndrew (topological). In particular, metric entropy, with the imposition of some additional properties, is proven to...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2019-09-01
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| Series: | Entropy |
| Subjects: | |
| Online Access: | https://www.mdpi.com/1099-4300/21/10/938 |
| _version_ | 1811187005144432640 |
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| author | Raymond Addabbo Denis Blackmore |
| author_facet | Raymond Addabbo Denis Blackmore |
| author_sort | Raymond Addabbo |
| collection | DOAJ |
| description | A rigorous dynamical systems-based hierarchy is established for the definitions of entropy of Shannon (information), Kolmogorov−Sinai (metric) and Adler, Konheim & 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. |
| first_indexed | 2024-04-11T13:54:55Z |
| format | Article |
| id | doaj.art-5d79768719654f0a80f323795cf38eb8 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-04-11T13:54:55Z |
| publishDate | 2019-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-5d79768719654f0a80f323795cf38eb82022-12-22T04:20:22ZengMDPI AGEntropy1099-43002019-09-01211093810.3390/e21100938e21100938A Dynamical Systems-Based Hierarchy for Shannon, Metric and Topological EntropyRaymond Addabbo0Denis Blackmore1Department of Arts and Sciences, Vaughn College of Aeronautics and Technology, Flushing, NY 11369, USADepartment of Mathematical Sciences and Center for Applied and Computational Mathematics, New Jersey Institute of Technology, Newark, NJ 07102-1982, USAA rigorous dynamical systems-based hierarchy is established for the definitions of entropy of Shannon (information), Kolmogorov−Sinai (metric) and Adler, Konheim & 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.https://www.mdpi.com/1099-4300/21/10/938topological entropyshannon entropy: metric entropybernoulli scheme |
| spellingShingle | Raymond Addabbo Denis Blackmore A Dynamical Systems-Based Hierarchy for Shannon, Metric and Topological Entropy Entropy topological entropy shannon entropy: metric entropy bernoulli scheme |
| title | A Dynamical Systems-Based Hierarchy for Shannon, Metric and Topological Entropy |
| title_full | A Dynamical Systems-Based Hierarchy for Shannon, Metric and Topological Entropy |
| title_fullStr | A Dynamical Systems-Based Hierarchy for Shannon, Metric and Topological Entropy |
| title_full_unstemmed | A Dynamical Systems-Based Hierarchy for Shannon, Metric and Topological Entropy |
| title_short | A Dynamical Systems-Based Hierarchy for Shannon, Metric and Topological Entropy |
| title_sort | dynamical systems based hierarchy for shannon metric and topological entropy |
| topic | topological entropy shannon entropy: metric entropy bernoulli scheme |
| url | https://www.mdpi.com/1099-4300/21/10/938 |
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