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A Dynamical Systems-Based Hierarchy for Shannon, Metric and Topological Entropy

1
Department of Arts and Sciences, Vaughn College of Aeronautics and Technology, Flushing, NY 11369, USA
2
Department of Mathematical Sciences and Center for Applied and Computational Mathematics, New Jersey Institute of Technology, Newark, NJ 07102-1982, USA
*
Author to whom correspondence should be addressed.
Entropy 2019, 21(10), 938; https://doi.org/10.3390/e21100938
Received: 12 August 2019 / Revised: 12 September 2019 / Accepted: 20 September 2019 / Published: 25 September 2019
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. View Full-Text
Keywords: topological entropy; Shannon entropy: metric entropy; Bernoulli scheme topological entropy; Shannon entropy: metric entropy; Bernoulli scheme
MDPI and ACS Style

Addabbo, R.; Blackmore, D. A Dynamical Systems-Based Hierarchy for Shannon, Metric and Topological Entropy. Entropy 2019, 21, 938.

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