6 - More about entropy

Summary

This chapter treats several topics in entropy theory that are somewhat beyond the basics. We begin by computing the entropies of automorphisms of the torus, skew products, and induced transformations. The following sections discuss convergence of the information per unit time (Shannon–McMillan–Breiman Theorem) and the topological version of entropy for cascades. We give an introduction to the Ornstein Isomorphism Theorem, which says that two Bernoulli schemes are isomorphic if and only if they have the same entropy. Ornstein's associated theory of sufficient conditions for m.p.t.s to be isomorphic to Bernoulli shifts has produced a surprising list of examples, including classical ones like geodesic maps and automorphisms of the torus, that are metrically indistinguishable from repeated independent random experiments. In the final section we present the Keane–Smorodinsky construction of the isomorphism whose existence is implied by Ornstein's theorem. Their work actually strengthens Ornstein's result, since they are able to construct the isomorphism explicitly, and the map is finitary: each coordinate of the image of a point can be calculated from knowledge of only a finite piece of the history of that point. (Alternatively, the map is a homeomorphism once a set of measure 0 has been deleted). This means that in principle such a coding can actually be carried out mechanically.