 In order to introduce the concept of ergodicity we have to introduce the phase and the time mean of a function \(f\) on \(X\). Mathematically speaking, a function assigns each point in \(X\) a number. If the numbers are always real the function is a real-valued function; and if the numbers may be complex, then it is a complex-valued function. Intuitively we can think of these numbers as representing the physical quantities of interest. Recalling the example of the bouncing ball, \(f\) could for instance assign each point in the phase space \(X\) the kinetic energy the system has at that point; in this case we would have \(f = p^2 / 2m\), where \(m\) is the mass of the ball. For every function we can take two kinds of averages. The first is the infinite time average \(f^*\). The general idea of a time average is familiar from everyday contexts. You play the lottery on three consecutive Saturdays. On the first you win $10; on the second you win nothing; and on the third you win $50. Your average gain is ($10 + $0 + $50)/3 = $20. Technically speaking this is a time average. This simple idea can easily be put to use in a dynamical system: follow the system’s evolution over time (and remember that we are now talking about an average for discrete points of time), take the value of the relevant function at each step, add the values, and then divide by the number of steps. This yields