In this chapter we look at two related notions that are important parameters for chaotic dynamical systems. The first is the fractal dimension of a set. By permitting noninteger values, this notion extends the topological concept of dimension to sets such as Cantor sets. While all Cantor sets are homeomorphic, they may look thicker or thinner depending on the parameters in their construction. Fractal dimension is a measure of the thickness of these sets. When the Cantor set in question arises as an invariant set of a hyperbolic dynamical system its dimension is related in deep ways to other dynamically important quantities, notably the contraction and expansion rates in the system. This is an active research topic, and we illustrate it with the Smale horseshoe.

The other notion is entropy. It measures the global orbit complexity on an exponential scale and is intimately related to the growth rate of periodic points and contraction and expansion rates. As an invariant of topological conjugacy, it also provides a means for telling apart dynamical systems that are not conjugate.

The values of dimension and entropy of an invariant set of a dynamical system are related, and so are the constructions involved in defining them. The common root is the notion of capacity of a set, with which we begin the chapter.