The notion of isomorphism in the category of measure preserving systems (defined, for instance, in this earlier post) is fairly flexible, as it allows measure perturbations. For instance, the system where is the Lebesgue measure and is isomorphic to the Bernoulli system , where is the left shift and is the infinite product of the measure on defined so that .
However, this flexibility can make it quite difficult to decide whether two given systems are isomorphic or not. While several spectral invariants can be used, such as ergodicity, weakly mixing or strongly mixing, they are not sufficient to distinguish even between two Bernoulli shifts. To address this problem, Kolmogorov and Sinai developed the notion of entropy in the late 1950’s. Since then, the entropy of measure preserving systems, together with its sister in topological dynamics, has became a fundamental tool in understanding the structure of dynamical systems, with applications far beyond distinguishing non-isomorphic systems.
In this post I will define and then briefly survey some of the basic properties of entropy. I’ll mostly follow the exposition in the upcoming book of Einsiedler, Lindenstrauss and Ward, whose draft is available here; other good sources are Walter’s book and, in a somewhat different perspective, Austin’s recent notes.