 Since \(\phi_t\) tells us for every point in \(X\) how it evolves in time, it also tells us how sets of points move around. For instance, choose an arbitrary set \(A\) in \(X\); then \(\phi_t (A)\) is the image of A after \(t\) time units under the dynamics of the system. This is illustrated in Figure 2b. Considering sets of points rather than single points is important when we think about physical applications of this mathematical formalism. We can never determine the exact initial condition of a ball bouncing on a spring. No matter how precisely we measure \(\gamma_0\), there will always be some measurement error. So what we really want to know in practical applications is not how a precise mathematical point evolves, but rather how a set of points around the initial condition \(\gamma_0\) evolves. In our example with the ball the evolution is ‘tame’, in that the set keeps its original shape. As we will see below, this is not always the case.