 Since we are free to choose \(A\) as we wish, we immediately get another important result: a system can be ergodic only if its trajectory may access all parts of \(X\) of positive measure, i.e., if the trajectory passes arbitrarily close to any point in \(X\) infinitely many times as time tends towards infinity. And this implies that the phase space of ergodic systems is called metrically indecomposable (or also ‘irreducible’ or ‘inseparable’): every set invariant under \(T\) (i.e., every set that is mapped onto itself under \(T)\) has either measure 0 or 1. As a consequence, \(X\) cannot be divided into two or more subspaces (of non-zero measure) that are invariant under \(T\). Conversely, a non-ergodic system is metrically decomposable. Hence, metric indecomposability and ergodicity are equivalent. A metrically decomposable system is schematically illustrated in Figure 3.