Entropy of infinite systems and transformations

Abstract

The Kolmogorov-Sinai entropy is a far reaching dynamical generalization of Shannon entropy of information systems. This entropy works perfectly for probability measure preserving (p.m.p.) transformations. However, it is not useful when there is no finite invariant measure. There are certain successful extensions of the notion of entropy to infinite measure spaces, or transformations with infinite invariant measures. The three main extensions are Parry, Krengel, and Poisson entropies. In this survey, we shortly overview the history of entropy, discuss the pioneering notions of Shannon and later contributions of Kolmogorov and Sinai, and discuss in somewhat more details the extensions to infinite systems. We compare and contrast these entropies with each other and with the entropy on finite systems.