Skip to content
DraconiticResearch
Search Sources API
203,203 studies31 sourcesUpdated 38 min ago
PreprintarXiv licencearXiv

A Multidimensional Birkhoff Theorem for some C0 Lagrangians

  • Skander Charfi
  • Ibrahim Trifa
Open at sourcePDFDOICite

Abstract

We prove a multidimensional Birkhoff theorem for a new class of "C0 Lagrangian subsets" in cotangent bundles, obtained as Hausdorff limits of compact exact Lagrangian submanifolds with control on their Liouville primitives. If the successive images of such a subset L under the flow of a Tonelli Hamiltonian admit convergent subsequences in both positive and negative time, then L and all its images are Lipschitz graphs over the base. This extends Birkhoff's celebrated theorem for twist maps of the annulus and its known higher-dimensional generalizations, and provides a first version of such results for merely continuous objects. The proof combines Floer-theoretic graph selectors, variational solutions of the Hamilton-Jacobi equation, and weak KAM theory. We also investigate the rigidity and uniqueness of limiting primitives for this new class of singular Lagrangian subsets, which may be of independent interest in C0 symplectic topology.

Related

  1. Birkhoff interpolation models for optimization with some available derivatives
    Jeffrey Larson, Matt Menickelly, Evan Toler
    arXivPreprintarXiv licenceFull textOptimisation & controlarXiv
  2. Wolff Potential and Birkhoff's Ergodic Theorem in Grand Variable Herz Spaces
    Rosalio G. Artes, Waqar Afzal, Samra Asjad +2
    Gulf Journal of MathematicsPeer-reviewedLicence not statedFull text7 citedGeneral mathematicsOpenAlex
  3. A Bernstein Theorem for the Self-Shrinking J-Equation and Some Generalizations
    Yiyang Pan, Wenlong Wang
    arXivPreprintarXiv licenceFull textAnalysis & PDEsGeometry & topologyarXiv
  4. Lower bounds for some value sets over finite fields: incidence geometry and Bourgain's group expansion theorem
    Xiyu Hu
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  5. Multidimensional Szemerédi Theorem for Pairwise Independent Jointly Intersective Polynomials
    Nikos Frantzikinakis, Davide Sclosa
    arXivPreprintarXiv licenceFull textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{charfi2026multidimensional,
  title = {{A Multidimensional Birkhoff Theorem for some $C^0$ Lagrangians}},
  author = {Skander Charfi and Ibrahim Trifa},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.28373},
  eprint = {2609.28373},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28373},
}
RIS
TY  - UNPB
TI  - A Multidimensional Birkhoff Theorem for some C^0 Lagrangians
AU  - Skander Charfi
AU  - Ibrahim Trifa
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.28373
UR  - https://arxiv.org/abs/2609.28373
AB  - We prove a multidimensional Birkhoff theorem for a new class of "C^0 Lagrangian subsets" in cotangent bundles, obtained as Hausdorff limits of compact exact Lagrangian submanifolds with control on their Liouville primitives. If the successive images of such a subset L under the flow of a Tonelli Hamiltonian admit convergent subsequences in both positive and negative time, then L and all its images are Lipschitz graphs over the base. This extends Birkhoff's celebrated theorem for twist maps of the annulus and its known higher-dimensional generalizations, and provides a first version of such results for merely continuous objects. The proof combines Floer-theoretic graph selectors, variational solutions of the Hamilton-Jacobi equation, and weak KAM theory. We also investigate the rigidity and uniqueness of limiting primitives for this new class of singular Lagrangian subsets, which may be of independent interest in C^0 symplectic topology.
ER  -