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.

Financial support: The visit of Alfonso Sorrentino is supported by the Simons Foundation (grant No. 615793). The event is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, agreement no. 075-15-2019-1614).

Organizer
Sorrentino Alfonso, Università degli Studi di Roma — Tor Vergata


Institutions
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center