In these lectures we discuss John Mather's variational approach to the study of convex and superlinear Hamiltonian systems, what is generally called Aubry-Mather theory. Starting from the observation that invariant Lagrangian graphs can be characterised in terms of their "action-minimizing" properties, we shall describe how analogue features can be traced in a more general setting, namely the so-called Tonelli Hamiltonian systems. This approach brings to light a plethora of compact invariant subsets for the system, which, under many points of view, can be seen as a generalisation of invariant Lagrangian graphs, despite not being in general either submanifolds or regular.

Besides being very significant from a dynamical systems point of view, these objects also appear in the study of weak solutions of the Hamilton-Jacobi equation (weak KAM theory) and play, as well, an important role in other different contexts: such as analysis, geometry, mathematical physics, billiard dynamics, etc. We shall also see how similar results can be also extended to some non-conservative setting, namely the case of so-called conformally symplectic systems.

Tentative course content:

• From KAM theory to Aubry-Mather theory: action-minimizing properties of invariant Lagrangian graphs.
• Tonelli Lagrangian and Hamiltonian on compact manifolds.
• Mather theory: Action-minimizing invariant measures, Mather sets and minimal average actions.
• Weak KAM theory: Hamilton-Jacobi equation, weak (sub)solutions, action-minimizing curves, Aubry sets and Mane sets.
• Aubry-Mather theory for conformally symplectic systems.

Some References:

• S. Maro', A. Sorrentino: "Aubry-Mather theory for conformally symplectic systems" Comm. Math. Phys., 354 (2): 775-808, 2017.
• A. Sorrentino: "Action-Minimizing Methods in Hamiltonian Dynamics. An Introduction to Aubry-Mather Theory". Mathematical Notes Series Vol. 50 (Princeton University Press), 2015.

Финансовая поддержка: Визит Альфонсо Соррентино в МИАН поддержан Фондом Саймонса (грант No. 615793). Мероприятие проводится при финансовой поддержке Минобрнауки России (грант на создание и развитие МЦМУ МИАН, соглашение № 075-15-2019-1614).

Руководитель
Соррентино Альфонсо

Организации
Математический институт им. В.А. Стеклова Российской академии наук, г. Москва
Математический центр мирового уровня «Математический институт им. В.А. Стеклова Российской академии наук» (МЦМУ МИАН)