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

Olga Bernardi, Francesco Morabito

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. Humilière and recently revisited by C. Viterbo. We establish a number of basic properties of this completion, in particular through the notion of $γ$-support, which we develop as a refinement of Humilière'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 Humilière 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.

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

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. Humilière and recently revisited by C. Viterbo. We establish a number of basic properties of this completion, in particular through the notion of -support, which we develop as a refinement of Humilière'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 Humilière 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.
Paper Structure (15 sections, 33 theorems, 165 equations, 10 figures)

This paper contains 15 sections, 33 theorems, 165 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Notations
  3. Preliminaries
  4. Liouville exact symplectic manifolds
  5. CS vector fields and diffeomorphisms
  6. Lagrangian submanifolds and Lagrangian branes
  7. Generating functions
  8. Spectral invariants
  9. Calculus of critical values
  10. Spectral invariants for Lagrangians
  11. Completions, gamma-support and gamma-coisotropy
  12. c-support, gamma-support and gamma-coisotropy
  13. Higher dimensional Birkhoff attractors
  14. Classical Birkhoff attractors
  15. Concluding remarks

Key Result

Proposition 2.8

Let $\varphi\in\mathrm{DHam}_c(M)$ be $C^1$--close to the identity. Then is the graph of an exact $1$-form on $\Delta$.

Figures (10)

  • Figure 1: An open Liouville manifold $W = M \cup_{\partial M \times \{0\}} (\partial M \times \mathbb{R}_{\geq 0})$.
  • Figure 2: On the left, an exact Lagrangian in $(T^*\mathbb{S}, -pdq)$: blue and red areas sum to zero. On the right, a non-essential curve surrounding a non-zero area.
  • Figure 3: $f(x) = \arctan x$ is not PS for the Euclidean metric on $\mathbb{R}^2$.
  • Figure 4: The vector field $X$, transverse to the level hypersurfaces of $f$.
  • Figure 5: The set $V$ is $\gamma$-coisotropic at $z \in V$ if $\gamma(\varphi)>\delta$.
  • ...and 5 more figures

Theorems & Definitions (96)

  • Remark 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Example 1
  • Example 2
  • Example 3
  • Remark 2.7
  • ...and 86 more