Higher Dimensional Birkhoff attractors (with an appendix by Maxime Zavidovique)

TL;DR

This paper generalizes Birkhoff attractors to higher dimensions, relates them to dissipative Hamilton-Jacobi systems, and explores their properties via $b3$-supports and Lagrangian limits, with counterexamples in the appendix.

Contribution

It introduces a higher-dimensional notion of Birkhoff attractors, proves their equivalence with classical ones, and analyzes their behavior in dissipative Hamilton-Jacobi and Lagrangian contexts.

Findings

Higher-dimensional Birkhoff attractors coincide with classical ones.

Solutions to discounted Hamilton-Jacobi equations are contained in the attractor.

The $b3$-support of a Lagrangian reflects the base's cohomology.

Abstract

We extend to higher dimensions the notion of Birkhoff attractor of a dissipative map. We prove that this notion coincides with the classical Birkhoff attractor. We prove that for the dissipative system associated to the discounted Hamilton-Jacobi equation the graph of a solution is contained in the Birkhoff attractor. We also study what happens when we perturb a Hamiltonian system to make it dissipative and let the perturbation go to zero. The paper contains two important results on $\gamma$-supports and elements of the $\gamma$-completion of the space of exact Lagrangians. Firstly the $\gamma$-support of a Lagrangian in a cotangent bundle carries the cohomology of the base and secondly given an exact Lagrangian $L$, any Floer theoretic equivalent Lagrangian is the $\gamma$-limit of Hamiltonian images of $L$. The appendix provides instructive counter-examples.