Entropy and Sinai’s Theorem

These short notes cover the basic definitions and properties of entropy, with an emphasis on its use in dynamical systems. They are intended to show how the different notions of entropy that appear in probability, mathematical statistical physics can be modified for use in the study of measurable dynamical systems, culminating in one of the most striking such uses of entropy: the proof of Sinai’s Theorem that entropic constraints are enough to find all Bernoulli factors of an ergodic invertible probability measure-preserving system. Our presentation loosely follows that in Smorodinsky’s monograph [7].