 The essential elements in the discussion so far were the phase space \(X\), the time evolution \(\phi_t\), and the measure \(\mu\). And these are also the ingredients for the definition of an abstract dynamical system. An abstract dynamical system is a triple \([X, \mu , T_t]\), where \(\{T_t \mid t \text{ is an instant of time}\}\) is a family of automorphisms, i.e., a family of transformations of \(X\) onto itself with the property that \(T_{t_1 +t_2} = T_{t_1}(T_{t_2})\) for all \(x \in X\) (Arnold and Avez 1968, 1); we say more about time  below.[2]  In the above example \(X\) is the phase space of the ball’s motion, \(\mu\) is the Lebesgue measure, and \(T_t\) is \(\phi_t\).