Entropy of dynamical systems

The word entropy is used exclusively to refer to the entropy of a dynamical system, i.e. a map or a flow. It measures the rate of increase in dynamical complexity as the system evolves with time. This is not to be confused with other notions of entropy connected with spatial complexity. I will attempt to give a brief survey of the role of entropy in dynamical systems and especially in smooth ergodic theory. The topics are chosen to give a flavor of this invariant; they are also somewhat biased toward my own interests. This article is aimed at nonexperts. After a review of some basic definitions and ideas, I will focus on one main topic, namely the relation of entropy to Lyapunov exponents and dimension, which I will discuss in some depth. Several other interpretations of entropy, including its relations to volume growth, periodic orbits and horseshoes, large deviations and rates of escape are treated briefly.