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

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.

💡 Research Summary

The notes, based on C. Viterbo’s lectures at the 2025 CIME School, develop a systematic theory of completing the space of exact Lagrangian submanifolds in a Liouville symplectic manifold with respect to the spectral (γ) metric introduced by V. Humilière. The authors first recall the basic objects: a Liouville manifold (M, −dλ), exact Lagrangians L⊂M, and Lagrangian branes (L, fL) where λ|L = dfL. They introduce the set L0(T* N) of exact Lagrangians Hamiltonian‑isotopic to the zero section of a cotangent bundle, and endow it with the γ‑distance, defined as the minimal action difference between two Lagrangians.

A central contribution is the construction of the metric completion cL0(T* N). To describe limits of Cauchy sequences, the authors refine Humilière’s notion of support into the γ‑support, a compact (or possibly non‑compact) subset of the cotangent bundle that records where the sequence “concentrates”. When the γ‑support is compact, the authors define a subspace cLc(T* N) consisting of those limit objects with compact γ‑support and prove that this subspace is itself complete. They establish basic properties such as the triangle inequality, non‑symmetry, and the fact that γ‑distance vanishes only for Hamiltonian‑isotopic Lagrangians. Moreover, they show that on cLc the γ‑metric coincides with the c‑metric (the metric induced by the generating‑function norm).

Generating‑function theory plays a crucial role. Every exact Lagrangian in a cotangent bundle can be represented by a generating function S(q, ξ) with auxiliary parameters ξ. The authors recall the definition of a generating function (regularity of ∂ξS) and the associated critical set ΣS. The image iS(ΣS) is an exact Lagrangian LS, and the correspondence LS∩ON ↔ Crit(S) generalises the familiar identification of graphs of exact 1‑forms with critical points. Two classes of generating functions are discussed: quadratic at infinity (GFQI) and asymptotically quadratic. Lemma 2.15 shows that any asymptotically quadratic generating function can be replaced by a GFQI without changing the associated Lagrangian, which guarantees stability under product operations.

The second part of the paper applies the completed Lagrangian space to conformally symplectic dynamics. A vector field Y satisfying LY ω = α ω (α∈ℝ) is called conformally symplectic (CS); its flow ψt rescales the symplectic form by e^{αt}. The authors prove that γ‑support is invariant under CS diffeomorphisms, which allows them to define a generalized Birkhoff attractor for a CS map ψ as

 B∞(ψ) = ⋂_{k≥0} ψ^{k}(cLc(T* N)).

When ψ is a genuine symplectomorphism (α=0) this reduces to the classical Birkhoff attractor B(ψ) studied by Birkhoff, Charpentier, and Le Calvez. For α≠0 the set B∞(ψ) retains the essential dynamical features: it is compact, invariant, attracts all points in a prescribed γ‑neighbourhood, and contains the minimal invariant set. The authors illustrate the construction with a damped Hamiltonian system (kinetic energy plus potential plus linear friction) and verify that the associated flow is CS with positive conformal rate.

Finally, the notes sketch several further directions. They mention possible relations between the γ‑completion and other symplectic metrics such as Hofer’s distance, the role of the completed space in Floer‑theoretic invariants, and the challenge of understanding non‑compact γ‑supports. Open questions include: (i) whether cL0 admits a natural smooth structure compatible with the γ‑metric; (ii) how spectral invariants behave under limits in the completion; (iii) extensions to non‑exact or non‑Liouville settings; and (iv) the existence of “spectral attractors” for broader classes of conformally symplectic systems.

In summary, the paper provides a rigorous foundation for the metric completion of exact Lagrangians, introduces the refined γ‑support concept, and demonstrates its power by constructing a generalized Birkhoff attractor for conformally symplectic dynamics, thereby opening new avenues at the interface of symplectic topology and dynamical systems.