Entropy of innite systems and transformations

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 nite invariant measure. There are certain successful extensions of the notion of entropy to innite measure spaces, or transformations with innite 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 innite systems. We compare and contrast these entropies with each other and with the entropy on nite systems.