Entropy in dynamical systems

In this chapter, the word entropy is used exclusively to refer to the entropy of a dynamical system. 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.