We have seen two ways in which uncertainty (and thus probability) may appear in the study of strictly deterministic systems. The first was the consequence of following a random distribution of initial states, which, in turn, led to a development of the notion of the Frobenius–Perron operator and an examination of its properties as a means of studying the asymptotic properties of flows of densities. The second resulted from the random application of a transformation S to a system and led naturally to our study of the linear Boltzmann equation.

In this chapter we consider yet another source of probabilistic distributions in deterministic systems. Specifically, we examine discrete time situations in which at each time the value xn+1 = S(xn) is reached with some error. An extremely interesting situation occurs when this error is small and the system is “primarily” governed by a deterministic transformation S. We consider two possible ways in which this error might be small: Either the error occurs rather rarely and is thus small on the average, or the error occurs constantly but is small in magnitude. In both cases, we consider the situation in which the error is independent of S(xn) and are, thus, led to first recall the notion of independent random variables in the next section and to explore some of their properties in Sections 10.2 and 10.3.