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.
All Science Journal Classification (ASJC) codes
- Physics and Astronomy(all)
- Bernoulli scheme
- Shannon entropy: metric entropy
- Topological entropy