Gibbs Measures and Ergodic Properties of Axiom A Diffeomorphisms

This work focuses on the study of Gibbs measures from two different points of view: the first probabilistic and the second physical-mathematical. Gibbs measures play a key role in the analysis of multidimensional lattice models and in the investigations of the ergodic properties of Axiom-A diffeomorphisms. As a theoretical foundation, we present two classic and general results concerning the local structure of a dynamical system, they are the Hartman-Grobman Theorem and the Stable Manifold Theorem. The work proceeds with an analysis of the theory of specifications and compatible measures on multidimensional lattices. Gibbs measures are presented as probability distributions compatible with specifications derived from the Boltzmann-Gibbs formula. We then turn our attention to the Axiom-A diffeomorphisms and we investigate their ergodic properties. The construction of an equilibrium state associated with a potential is analyzed, and the reference texts for this part are the original works of Bowen, Ruelle and Sinai.