ABSTRACT
Since its introduction into ergodic theory, entropy has played a central role in the study of dynamical systems, with wide-ranging applications to fields such as number theory, smooth dynamics, and operator algebras. This chapter provides some additional context and motivation for readers who have some familiarity with operator algebras, where the ideas of external and internal approximation that account for the conceptual distinction between amenability and soficity have long played an instrumental role in elucidating structure and establishing classification theorems. It reviews the Kolmogorov-Sinai approach to entropy in its structurally most general setting of amenable groups. The chapter introduces the basic theory of sofic measure entropy and sofic topological entropy and discusses a dual approach to sofic measure entropy which brings it into closer technical alignment with sofic topological entropy. The chapter presents a recent theorem of Hayes which gives a formula for the sofic entropy of principal algebraic actions in terms of the FugledeKadison determinant.
