 So far we have described \(T_t\) as giving the time evolution of a system. Now let us look at this from a more mathematical point of view: the effect of \(T_t\) is that it assigns to every point in \(X\) another point in \(X\) after \(t\) time units have elapsed. In the above example \(\gamma_1\) is mapped onto \(\gamma_2\) under \(\phi_t\) after \(t = 5\) seconds. Hence, from a mathematical point of view the time evolution of a system consists in a mapping of \(X\) onto itself, which is why the above definition takes \(T_t\) to be a family of mappings of \(X\) onto itself. Such a mapping is a prescription that tells you for every point \(x\) in \(X\) on which other point in \(X\) it is mapped (from now on we use \(x\) to denote any point in \(X\), and it no longer stands, as in the above example, for the position of the ball). 