Aubry Set on Infinite Cyclic Coverings

IF 0.8 4区数学 Q3 MATHEMATICS, APPLIED

Regular and Chaotic Dynamics Pub Date : 2023-07-31 DOI:10.1134/S1560354723520015

Albert Fathi, Pierre Pageault

引用次数: 0

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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无限循环覆盖上的Aubry集

在本文中，我们研究了定义在紧致流形（M）的切丛上的TonelliLagrangian（L）到\（M）无限循环覆盖的提升的投影Aubry集。大多数弱KAM和Aubry-Mather理论都可以在这种情况下完成。我们给出了提升拉格朗日的投影Aubry集为空的一个充要条件，该集同时涉及\（L\）的Mather最小化测度和Mather类。最后，我们给出了二维环面上的Mañè例子，表明当覆盖不是无限循环时，我们的结果不一定成立。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Regular and Chaotic Dynamics 物理-力学

CiteScore

2.50

自引率

7.10%

发文量

审稿时长

>12 weeks

期刊介绍： Regular and Chaotic Dynamics (RCD) is an international journal publishing original research papers in dynamical systems theory and its applications. Rooted in the Moscow school of mathematics and mechanics, the journal successfully combines classical problems, modern mathematical techniques and breakthroughs in the field. Regular and Chaotic Dynamics welcomes papers that establish original results, characterized by rigorous mathematical settings and proofs, and that also address practical problems. In addition to research papers, the journal publishes review articles, historical and polemical essays, and translations of works by influential scientists of past centuries, previously unavailable in English. Along with regular issues, RCD also publishes special issues devoted to particular topics and events in the world of dynamical systems.