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This chapter discusses the notion of action-minimizing orbits. In particular, it defines the other two families of invariant sets, the so-called Aubry and Mañé sets. It explains their main dynamical and symplectic properties, comparing them with the results obtained in the preceding chapter for the Mather sets. The relation between these new invariant sets and the Mather sets is described. As a by-product, the chapter introduces the Mañé's potential, Peierls' barrier, and Mañé's critical value. It discusses their properties thoroughly. In particular, it highlights how this critical value is related to the minimal average action and describes these new concepts in the case of the simple pendulum.

This chapter defines cohomology of Aubry-Mather type and explains why it implies one of the diffusion mechanisms, after a generic perturbation. The definition of Aubry-Mather type includes a much simpler case, that is when the Aubry set is a hyperbolic periodic orbit, still contained in a normally hyperbolic invariant cylinder. This definition says that each of the two local components of the Aubry set is of Aubry-Mather type. There is another type of bifurcation in which one component of the Aubry set is of Aubry-Mather type with an invariant cylinder and another is a hyperbolic periodic orbit. This can be called the asymmetric bifurcation. This case appears at double resonance, when the shortest loop is simple non-critical.

This chapter describes forcing relations, different diffusion mechanisms, and Aubry-Mather types. The Aubry set can be decomposed into disjoint invariant sets called static classes, which gives important insight into the structure of the Aubry set. The chapter then formulates Theorem 2.2 and shows that it implies the book's main theorem. It utilizes the concept of forcing equivalence. The actual definition is not important for the current discussions, instead, the chapter states its main application to Arnold diffusion. The chapter also looks at symplectic coordinate changes. The definition of exact symplectic coordinate change for a time-periodic system is somewhat restrictive, and in particular, it does not apply directly to the linear coordinate change performed at the double resonance.

This chapter proves Aubry-Mather type in the single-resonance regime. It proves that for a single-resonance normal form system which satisfies the non-degeneracy conditions, every c in the resonance curve is of either Aubry-Mather or bifurcation Aubry-Mather type. The main results are Theorems 9.3 and 9.5, which restate Propositions 3.9 and 3.10. The chapter then proves that the conditions hold on an open and dense set of Hamiltonians. It discusses bifurcations in the double maxima case, as well as hyperbolic coordinates. The chapter also examines normally hyperbolic invariant cylinder, the localization of the Aubry and Mañé sets, and the genericity of the single-resonance conditions.

This chapter proves Aubry-Mather type for the double resonance regime. It begins by considering the “non-critical energy case” and showing that the cohomologies as chosen are of Aubry-Mather type. The proof consists of two cases. In the first case, the chapter uses the almost verticality of the cylinder, and the idea is similar to the proof of Theorem 9.3. It applies the a priori Lipschitz estimates for the Aubry sets. In the second case, the chapter uses the strong Lipschitz estimate for the energy, and the idea is similar to the proof of Theorem 11.1. It then looks at the construction of the local coordinates. This is done separately near the hyperbolic fixed point (local) and away from it (global).