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

This paper, based on C. Viterbo's 2025 CIME lectures, establishes fundamental properties of the completion of Lagrangian submanifolds under the spectral metric via the refined concept of $\gamma$-support and applies these results to conformally symplectic dynamics to generalize the notion of Birkhoff attractors.

The Gist

Imagine you are standing in a vast, infinite library. This library represents the world of Symplectic Geometry, a branch of mathematics that studies how shapes move and interact in a special kind of space (like the phase space of physics, where position and momentum live together).

In this library, there are special, perfectly smooth sheets of paper called Lagrangian submanifolds. Think of these as "perfectly balanced" shapes. They have a very specific rule: they can't twist or stretch in a way that breaks the library's fundamental laws of motion.

For a long time, mathematicians could only study these sheets if they were perfectly smooth and well-behaved. But what happens if you take a smooth sheet, crumple it, tear it, or let it evolve under a chaotic force until it becomes a messy, jagged blob? Does it still belong in the library?

This paper is about building a completion of this library. It's like saying, "We will accept not just the perfect sheets, but also the messy, crumpled, limit versions of them." The authors introduce a new way to measure the "distance" between these shapes and show that even the messiest shapes have a structure we can understand.

Here is a breakdown of the paper's key ideas using everyday analogies:

1. The "Spectral Ruler" (Measuring Distance)

Imagine you have two different shapes in your library. How do you measure how far apart they are? You can't just use a tape measure because these shapes are flexible and can wiggle.

The authors use a tool called Spectral Invariants. Think of this like a "vibration sensor." If you pluck a guitar string (a shape), it vibrates at specific frequencies. The "spectral invariant" is a number that tells you the lowest or highest frequency of that vibration.

• The Analogy: If you have two different guitar strings, the difference in their "lowest note" tells you how different they are.
• The Magic: The authors proved that this "vibration distance" is a perfect ruler. It satisfies all the rules of a distance (it's positive, symmetric, and follows the triangle inequality). This allows them to treat the space of these shapes like a geometric landscape.

2. The "Completion" (Filling in the Gaps)

In math, a "complete" space is one where if you have a sequence of things getting closer and closer together, they eventually land on a specific, existing object.

• The Problem: If you take a sequence of smooth, perfect sheets and crumple them more and more, they might approach a shape that is not a smooth sheet anymore. It might be a fuzzy cloud or a jagged fractal. In the old library, this fuzzy cloud didn't exist; the sequence just "ran off the edge."
• The Solution: The authors built a completion. They said, "Okay, let's create a new section of the library for these fuzzy clouds." They call this the Humilière Completion. Now, every sequence that gets closer and closer has a home.

3. The "Shadow" (Gamma-Support)

Now that we have these fuzzy, completed shapes, how do we see them? They aren't smooth lines anymore, so we can't just draw them.

• The Analogy: Imagine a cloud. You can't see the exact edge of the cloud, but you can see its shadow on the ground. The shadow tells you where the cloud is and how big it is, even if the cloud itself is fluffy and undefined.
• The Concept: The authors define the $\gamma$-support. This is the "shadow" of the fuzzy shape. It's the set of points in space where the shape is "active."
• The Discovery: They proved that these shadows are never just random blobs. They always have a specific geometric property called $\gamma$-coisotropy.
  • Simple translation: Think of a coisotropic shape as a "thick" line or a "thick" sheet. It has a certain minimum thickness. The authors found that even the most chaotic, fuzzy shapes cast a shadow that is "thick" in a very specific mathematical way. You can't shrink the shadow to a single point; it always has some "bulk."

4. The "Birkhoff Attractor" (The Cosmic Dust Collector)

The second half of the paper applies this new library to Dynamics (how things move over time).

• The Scenario: Imagine a machine that takes a shape, shrinks it, and moves it (a "dissipative" system, like a pendulum slowing down due to friction).
• The Old View: In 2D (like a flat sheet of paper), mathematicians knew that these machines eventually settle down on a specific "attractor"—a final resting spot for the shape. This was called the Birkhoff Attractor.
• The New View: What happens in higher dimensions (3D, 4D, etc.)? The old tools couldn't handle the complexity.
• The Breakthrough: Using their new "fuzzy library" and the "shadow" concept, the authors defined a Generalized Birkhoff Attractor for any dimension.
  • The Analogy: Imagine a whirlpool in a river. In 2D, you can see the center clearly. In 3D, the water might be churning in a way that looks like a solid block of foam. The authors proved that even in this chaotic foam, there is a core "shadow" that the system always returns to. This shadow is the new attractor.

Why Does This Matter?

• It Unifies Chaos and Order: It gives mathematicians a way to talk about "messy" shapes as if they were "clean" objects.
• It Solves Old Mysteries: It helps answer questions about how complex systems (like weather patterns or planetary orbits with friction) settle down.
• It's a New Language: It provides a new vocabulary (spectral invariants, $\gamma$-support) to describe the "shape" of chaos.

Summary in One Sentence

The authors built a new mathematical "library" that accepts messy, crumpled shapes, invented a special ruler to measure them, and used this to prove that even in chaotic, high-dimensional systems, there is a hidden, thick "shadow" (an attractor) that acts as a permanent home for the motion.

Technical Summary

Based on the provided lecture notes, here is a detailed technical summary of the paper "The completion of the set of Lagrangians and applications to dynamics" by Olga Bernardi and Francesco Morabito, based on lectures by Claude Viterbo.

1. Problem Statement

The paper addresses the geometric and dynamical limitations of the space of exact Lagrangian submanifolds in a symplectic manifold, specifically within the cotangent bundle $T^*N$ of a closed smooth manifold $N$.

• The Gap: The space of exact Lagrangians isotopic to the zero section, denoted $\mathcal{L}_0(T^*N)$, equipped with the spectral metric $\gamma$ (introduced by Humilière and Viterbo), is not complete. Cauchy sequences of Lagrangians may converge to "singular" objects that are no longer smooth submanifolds.
• The Challenge: Understanding the geometric nature of these limit objects (the completion $\widehat{\mathcal{L}}_0(T^*N)$) and utilizing this completion to study dissipative (conformally symplectic) dynamical systems, particularly generalizing the concept of the Birkhoff attractor to higher dimensions.

2. Methodology

The authors employ a synthesis of symplectic topology, variational calculus, and dynamical systems theory:

• Generating Functions Quadratic at Infinity (GFQI): The core tool is the use of GFQIs to represent exact Lagrangians. The spectral invariants $c(\alpha, L)$ are defined via min-max critical values of these generating functions, utilizing Lusternik-Schnirelmann theory.
• Spectral Metrics: The paper utilizes two spectral distances:
  • $\gamma(L_1, L_2)$: A distance on the space of Lagrangians (modulo exact forms).
  • $c(\tilde{L}_1, \tilde{L}_2)$: A distance on the space of Lagrangian branes (Lagrangians with a specific primitive function).
• Completion and $\gamma$-Support: The authors construct the metric completions $\widehat{\mathcal{L}}_0(T^*N)$ and $\widehat{\mathcal{L}}_0(T^*N)$. To give geometric meaning to elements in these completions, they introduce the $\gamma$-support, defined as the set of points where the object cannot be displaced by a Hamiltonian diffeomorphism of arbitrarily small spectral norm.
• Conformally Symplectic Dynamics: The framework is applied to Conformally Exact Symplectic (CES) diffeomorphisms $\psi$ where $\psi^*\omega = a\omega$ with $0 < a < 1$. These maps act as contractions on the spectral metric space, allowing the application of the Banach Fixed Point Theorem.

3. Key Contributions and Results

A. Geometry of the Completion

• Existence of Limits: The paper establishes that the space of exact Lagrangians is not complete under the spectral metric. The completion contains "singular" Lagrangians that are not smooth submanifolds.
• $\gamma$-Support and $\gamma$-Coisotropy:
  • The authors define the $\gamma$-support ($\gamma\text{-supp}(L)$) of an element in the completion.
  • They prove that $\gamma$-supports are $\gamma$-coisotropic subsets. This generalizes the classical notion of coisotropic submanifolds. A set $V$ is $\gamma$-coisotropic if displacing it locally requires a non-zero amount of spectral energy.
  • Theorem 4.11: If $L \in \widehat{\mathcal{L}}_0(T^*N)$, then $\gamma\text{-supp}(L)$ is $\gamma$-coisotropic.
• Non-Triviality: For any $L$ in the completion with compact support, the $\gamma$-support is non-empty and intersects every vertical fiber $T^*_q N$.
• Dimensionality: The Hausdorff dimension of a $\gamma$-support is at least $n$ (the dimension of the base manifold). The paper discusses "Peano Lagrangians" where the support can be a space-filling curve or higher-dimensional sets.

B. Generalized Birkhoff Attractors

• Definition: For a CES diffeomorphism $\psi$ with conformal ratio $a \in (0,1)$, the map acts as a contraction on the complete metric space $\widehat{\mathcal{L}}_0(T^*N)$. By the Banach Fixed Point Theorem, there exists a unique fixed point $L_\infty$.
• The Attractor: The Generalized Birkhoff Attractor is defined as $B_\infty(\psi) := \gamma\text{-supp}(L_\infty)$.
• Properties:
  • $B_\infty$ is a compact, invariant, closed, and $\gamma$-coisotropic subset of $T^*N$.
  • Theorem 5.10: In the specific case of the cylinder ($N=S^1$), the generalized Birkhoff attractor $B_\infty$ coincides exactly with the classical topological Birkhoff attractor $B$ (defined as the intersection of the closures of the invariant regions separated by the attractor).
• Topological Complexity: The paper proves that the projection of $B_\infty$ onto the base $N$ induces an injective map on cohomology ($H^*(N) \to H^*(B_\infty)$), implying $B_\infty$ is topologically non-trivial.

C. Applications to Dynamics

• Dissipative Systems: The framework provides a robust tool for analyzing dissipative systems (e.g., damped pendulums, discounted Hamilton-Jacobi equations) in arbitrary dimensions, where classical invariant set theory often fails.
• Viscosity Solutions: The graph of the viscosity solution to the discounted Hamilton-Jacobi equation is shown to be contained within the generalized Birkhoff attractor.

4. Significance

• Unification of Analysis and Geometry: The work bridges the gap between the analytical study of Hamilton-Jacobi equations (via viscosity solutions) and symplectic topology (via Lagrangian intersections and spectral invariants).
• Higher-Dimensional Dynamics: It successfully generalizes the concept of the Birkhoff attractor from 2D annuli to higher-dimensional cotangent bundles, a problem that was previously intractable using classical topological methods.
• New Geometric Objects: The introduction of $\gamma$-coisotropic sets and the study of the completion $\widehat{\mathcal{L}}_0$ reveal a rich class of singular geometric objects that act as attractors in symplectic dynamics.
• Nearby Lagrangian Conjecture: The results provide a "weak" version of the Nearby Lagrangian Conjecture, showing that any two exact Lagrangians can be mapped to each other by an element in the completed group of Hamiltonian diffeomorphisms.

5. Open Questions Highlighted

The paper concludes by identifying several open problems:

• Connectivity: Is the generalized Birkhoff attractor $B_\infty$ always connected? (Proven for $T^*S^1$, open for higher dimensions).
• Dimensionality: What is the precise Hausdorff dimension of $\gamma$-supports? Can they be strictly larger than $n$?
• Uniqueness: For a given $\gamma$-support $V$, how many distinct elements in the completion share this support? The authors conjecture finiteness if $\dim_H(V) \leq n$.

In summary, this paper establishes a rigorous framework for the "completion" of Lagrangian submanifolds, using spectral invariants to define a geometry on singular objects, and demonstrates the profound utility of this framework in understanding the global structure of dissipative symplectic dynamics in high dimensions.