In this chapter we develop the ergodic theory of expanding maps on compact metric spaces. This theory evolved from the kind of ideas in statistical mechanics that we discussed in Section 10.3.4 and, for that reason, is often called thermodynamic formalism. We point out, however, that this last expression is much broader, encompassing not only the original setting of mathematical physics but also applications to other mathematical systems, such as the so-called uniformly hyperbolic diffeomorphisms and flows (in this latter regard, see the excellent monograph of Rufus Bowen [Bow75a]).

The main result in this chapter is the following theorem of David Ruelle, which we prove in Section 12.1 (the notion of Gibbs state is also introduced in Section 12.1):

Theorem 12.1 (Ruelle). Let f : M →M be a topologically exact expanding map on a compact metric space and φ : M → ℝ be a Hölder function. Then there exists a unique equilibrium state μ for φ. Moreover, the measure μ is exact, it is supported on the whole of M and is a Gibbs state.

Recall that an expanding map is topologically exact if (and only if) it is topologically mixing (Exercise 11.2.2). Moreover, a topologically exact map is necessarily surjective.

In the particular case when M is a Riemannian manifold and f is differentiable, the equilibrium state of the potential φ = −log | detDf | coincides with the absolutely continuous invariant measure given by Theorem 11.1.2. In particular, it is the unique physical measure of f. These facts are proved in Section 12.1.8.

The theorem of Livšic that we present in Section 12.2 complements the theorem of Ruelle in a very elegant way. It asserts that two potentials φ and ψ have the same equilibrium state if and only if the difference between them is cohomologous to a constant. In other words, this happens if and only if φ−ψ =c+uof −u for some c∈ℝ and some continuous function u.Moreover, and remarkably, it suffices to check this condition on the periodic orbits of f.

In Section 12.3 we show that the system (f,μ) exhibits exponential decay of correlations in the space of Hölder functions, for every equilibrium state μ of any Hölder potential.