 Let \(f\) be a function defined on some state space \(S\). In the continuous case, \(f\) would vary continuously on \(S\) and we might have a differential equation specifying how \(f\) varies. In the discrete case, \(f\) can be thought of as a mapping that can be iterated or reapplied a number of times. To indicate this, we can write \(f^{n}(x)\), meaning \(f\) is applied iteratively \(n\) times. For instance, \(f^{3}(x)\) would indicate \(f\) has been applied three times, thus \(f^{3}(x) = f(f(f(x)))\) (Robert May’s classic 1976 review article has a nice discussion of this for the logistic map, \(x_{n + 1} = rx_{n}(1 - x_{n})\), which arises in modeling the dynamics of predator-prey relations, for instance.). Furthermore, let \(K\) be a subset of \(S\).  Then \(f(K)\) represents \(f\) applied to the set of points \(K\), that is, \(f\) maps the set \(K\) into \(f(K)\). If \(f(K) = K\), then \(K\) is an invariant set under \(f\). 