In the course of our discussions of the baker's map, we noticed that we could easily use its isomorphism with the Bernoulli sequences to locate periodic orbits of the map. As we show below, we can exploit this isomorphism to prove that periodic orbits of the baker's map form a dense set in the unit square. Moreover, we will prove, without much difficulty, that the periodic orbits of the hyperbolic toral automorphisms are also dense in the unit square (or torus). A natural question to ask is: If these periodic orbits are ubiquitous, can they be put to some good use? In this chapter, we outline some simple affirmative answers to this question in the context of nonequilibrium statistical mechanics. In particular, we will see that periodic orbit expansions are natural objects when one encounters the need for the trace of a Frobenius–Perron operator, and when one wants to make explicit use of an (∈, T)-separated set. Moreover, the periodic orbits of a classical system form a natural starting point for a semi-classical version of quantum chaos theory. We should also mention that there is a new field of study dealing with issues related to the control of chaos, which exploits the presence of periodic orbits to slightly perturb a system from chaotic behavior to a more easily controlled periodic behavior.

Dense sets of unstable periodic orbits

Here we consider a hyperbolic system. If we have located a periodic orbit of our system, then each point on it has a set of stable and unstable directions.