Most of the considerations in this chapter are only valid for special classes of chaotic maps f, namely for either expanding or hyperbolic maps (for the definitions see section 15.6). We shall first derive a very important variational principle for the topological pressure. The notion of Gibbs measures and SRB measures (Sinai–Ruelle–Bowen measures) will be introduced. We shall then show that for one-dimensional expanding systems one type of free energy is sufficient: all interesting quantities such as the (dynamical) Rényi entropies, the generalized Liapunov exponents, and the Rényi dimensions can be derived from the topological pressure. A disadvantage is that, due to the expansion condition, we consider quite a restricted class of systems.

A variational principle for the topological pressure

Similar to the principle of minimum free energy in conventional statistical mechanics (see section 6.3) there is also a variational principle for the topological pressure. This variational principle allows us to distinguish the natural invariant measure of an expanding (or hyperbolic) map from other, less important invariant measures. In fact, the ‘physical meaning’ of the variational principle could be formulated as follows. Among all possible invariant measures of a map one is distinguished in the sense that it is the smoothest one along the unstable manifold.