The present monograph is based on lectures given in the mathematics departments of Berkeley (1973) and of Orsay (1974–1975). My aim has been to describe the mathematical structures underlying the thermodynamic formalism of equilibrium statistical mechanics, in the simplest case of classical lattice spin systems.

The thermodynamic formalism has its origins in physics, but it has now invaded topological dynamics and differentiable dynamical systems, with applications to questions as diverse as the study of invariant measures for an Anosov diffeomorphism (Sinai [3]), or the meromorphy of Selberg's zeta function (Ruelle [7]). The present text is an introduction to such questions, as well as to more traditional problems of statistical mechanics, like that of phase transitions. I have developed the general theory — which has considerable unity — in some detail. I have, however, left aside particular techniques (like that of correlation inequalities) which are important in discussing examples of phase transitions, but should be the object of a special study.

Statistical mechanics extends to systems vastly more general than the classical lattice spin systems discussed here (in particular to quantum systems). One can therefore predict that the theory discussed in this monograph will extend to vastly more general mathematical setups (in particular to non-commutative situations). I hope that the present text may contribute some inspiration to the construction of the more general theories, as well as clarifying the conceptual structure of the existing formalism.