Now we want to discuss a number of topics that are essential for an understanding of dynamical systems theory and which also play a role in a more detailed discussion of the relation between transport theory and dynamical systems theory. We begin with a discussion of the Kolmogorov-Sinai (KS) entropy, which is a characteristic property of those deterministic dynamical systems with ‘randomness’ properties similar to Bernoulli shifts discussed earlier. The KS entropy is essential for formulating the escape-rate expressions for transport coefficients, to be discussed in Chapters 11 and 12.

Heuristic considerations

Let us return for a moment to the Arnold cat map discussed in the previous chapter. There we illustrated an initial set, A, say, that is located in the lower left-hand corner of the unit square (see Fig. 8.3). As this set evolves under the action of the map TA, the set becomes longer and thinner so that after three iterations, the set has begun to fold back across the unit square, and after ten iterations, the set is so stretched out that it appears to cover the unit square uniformly (see Figs 8.5 and 8.6, respectively). Since the initial set A is getting stretched along the unstable direction, at every iteration we learn more about the initial location of the points within the initial set A That is, suppose that we can distinguish two points on the unit square only if they are separated by a distance δ, the resolution parameter, and suppose further that the characteristic dimension of the initial set, A, is of the order of δ.