 Discrete versus continuous time. We may consider discrete instants of time or a continuum of instants of time. For ease of presentation, we shall say in the first case that time is discrete and in the second case that time is continuous. This is just a convenient terminology that has no implications for whether time is fundamentally discrete or continuous. In the above example with the ball time was continuous (it was taken to be a real number). But often it is convenient to regard time as discrete. If time is continuous, then \(t\) is a real number and the family of automorphisms is \(\{T_t \mid t \in \mathbb{R} \}\), where \(\mathbb{R}\) is the set of real numbers. If time is discrete, then \(t\) is in the set \(\mathbb{Z} = \{\ldots -2, -1, 0, 1, 2, \ldots \}\), and the family of automorphisms is \(\{T_t \mid t \in \mathbb{Z}\}\). In order to indicate that we are dealing with a discrete family rather than a continuous one we sometimes replace ‘\(T_t\)’ with ‘\(T_n\)’; this is just a notational convention of no conceptual  importance.[3]  In such systems the progression from one instant of time to the next is also referred to as a ‘step’. In population biology, for instance, we often want to know how a population grows over a typical breeding time (e.g. one year). In mathematical models of such a population the points in \(X\) represent the size of a population (rather than the position and the momentum of a ball, as in the above example), and the transformation \(T_n\) represents the growth of the population after \(n\) time units. A simple example would be \(T_n = x + n\).