Abstract
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.
Highlights
Entropy, which can among a variety of other things, be roughly viewed as a measure of uncertainty, has been and remains a fascinating and still not completely understood concept with an amazing range of applications, including quantum and ecological systems [2]
Entropy 2019, 21, 938 we prove that metric entropy is a special case of topological entropy if one adds just a few assumptions
We shall prove there is a topology on numerous discrete measurable dynamical systems (DMDS) of interest such that the corresponding topological entropy is equal to the K–S
Summary
Entropy, which can among a variety of other things, be roughly viewed as a measure of uncertainty (cf. [1]), has been and remains a fascinating and still not completely understood concept with an amazing range of applications, including quantum and ecological systems [2]. There have been several extensive investigations of various types of entropy including historical accounts of the relevant developments and informative investigations of the linkages among the various forms of entropy such as in [3,4,5,6,7,8,9,10,11,12,13], but they all appear to be somewhat lacking in terms of identification of truly compelling unifying themes for the multifarious definitions Our intention in this and a subsequent paper is to provide a rigorous partial answer to this question by proving that there is a dynamical systems thread connecting all of the following definitions of entropy leading, with the possible addition of some mild assumptions, to the hierarchy. We begin with the topological entropy of a discrete dynamical system on a topological space
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