The completion of the set of Lagrangians and applications to dynamics -- Based on lectures by C. Viterbo

TL;DR

This paper introduces the completion of the set of Lagrangian submanifolds in symplectic manifolds using the spectral metric, explores its properties, and applies it to conformally symplectic dynamics, extending classical concepts like Birkhoff attractors.

Contribution

It develops the notion of the $ ext{γ}$-support within the completion of Lagrangian submanifolds and applies this framework to generalize Birkhoff attractors in symplectic dynamics.

Findings

Established basic properties of the spectral metric completion.

Developed the concept of $ ext{γ}$-support as a refinement.

Applied the framework to conformally symplectic dynamics and generalized Birkhoff attractors.

Abstract

The goal of these lectures is to introduce the completion of the set of Lagrangian submanifolds of a symplectic manifold with respect to the spectral metric first introduced by V. Humili\`ere and recently revisited by C. Viterbo. We establish a number of basic properties of this completion, in particular through the notion of $\gamma$-support, which we develop as a refinement of Humili\`ere's original concept. We then present an application of these notions to conformally symplectic dynamics, generalizing the notion of Birkhoff attractor as defined and studied by G.D. Birkhoff, M. Charpentier, and more recently P. Le Calvez. Finally, we briefly mention several other applications of the Humili\`ere completion and highlight many open questions. These are notes elaborated from the lectures with the same title given by C. Viterbo at the CIME School ''Symplectic Dynamics and Topology'' held in Cetraro (CS), Italy, from 16th to 20th June 2025.