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Aubry Set on Infinite Cyclic Coverings

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Abstract

In this paper, we study the projected Aubry set of a lift of a Tonelli Lagrangian \(L\) defined on the tangent bundle of a compact manifold \(M\) to an infinite cyclic covering of \(M\). Most of weak KAM and Aubry – Mather theory can be done in this setting. We give a necessary and sufficient condition for the emptiness of the projected Aubry set of the lifted Lagrangian involving both Mather minimizing measures and Mather classes of \(L\). Finally, we give Mañè examples on the two-dimensional torus showing that our results do not necessarily hold when the cover is not infinite cyclic.

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  • 29 October 2023

    The numbering issue has been changed to 4-5.

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Funding

This work was supported by ANR-12-BS01-0020 WKBHJ.

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Authors and Affiliations

  1. Georgia Institute of Technology & ENS de Lyon (Emeritus), School of Mathematics, 30332, Atlanta GA, USA

    Albert Fathi

  2. Lycée Etienne Mimard, 32 Rue Etienne Mimard, 42000, Saint-Étienne, France

    Pierre Pageault

Authors
  1. Albert Fathi
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  2. Pierre Pageault
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Correspondence to Albert Fathi or Pierre Pageault.

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The authors declare that they have no conflicts of interest.

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A Alain Chenciner pour ses 80 ans (et 50 ans d’interactions)!

MSC2010

37J50

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Fathi, A., Pageault, P. Aubry Set on Infinite Cyclic Coverings. Regul. Chaot. Dyn. 28, 425–446 (2023). https://doi.org/10.1134/S1560354723520015

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