Non-convex Mather's theory and the Conley conjecture on the cotangent bundle of the torus

Claude Viterbo

doi:10.3934/jmd.2025017

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2025, Volume 21: 719-753. Doi: 10.3934/jmd.2025017

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Non-convex Mather's theory and the Conley conjecture on the cotangent bundle of the torus

Claude Viterbo

Département de Mathématiques, Université de Paris-Sud, Paris-Saclay, F-91405 Orsay Cedex, France

Received: December 27, 2023

Revised: July 17, 2025

Published online: December 19, 2025

Part of this work was done while the author was at Centre de Mathématiques Laurent Schwartz, UMR 7640 du CNRS, École Polytechnique - 91128 Palaiseau, France and DMA, UMR 8553 du CNRS École Normale Supérieure-PSL University, 45 Rue d'Ulm, 75230 Paris Cedex 05, France. Supported also by Agence Nationale de la Recherche projects Symplexe, WKBHJ and MICROLOCAL.

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Abstract

The aim of this paper is to use the methods and results of symplectic homogenization (see [59]) to prove existence of periodic orbits and invariant measures with rotation vector belonging to the set of differentials of the Homogenized Hamiltonian. We also prove the Conley conjecture on the cotangent bundle of the torus. Both proofs rely on Symplectic Homogenization and a refinement of it.

Mathematics Subject Classification: Primary: 37J12; Secondary: 53D40.

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Figure 1. The Lagrangian $ L $ and the sets $ {{\rm{LConv}}_{N}}(L) $ and $ {\rm Conv}_{N}(L) $. For $ {{\rm{LConv}}_{N}}(L) $, the two "ears" have equal area

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