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Generic Properties of Mañé’s Set of Exact Magnetic Lagrangians

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Abstract

Let \(M\) be a closed manifold and \(L\) an exact magnetic Lagrangian. In this paper we prove that there exists a residual set \(\mathcal{G}\) of \(H^{1}\left(M;\mathbb{R}\right)\) such that the property

$${\widetilde{\mathcal{M}}}\left(c\right)={\widetilde{\mathcal{A}}}\left(c\right)={\widetilde{\mathcal{N}}}\left(c\right),\forall c\in\mathcal{G},$$

with \({\widetilde{\mathcal{M}}}\left(c\right)\) supporting a uniquely ergodic measure, is generic in the family of exact magnetic Lagrangians. We also prove that, for a fixed cohomology class \(c\), there exists a residual set of exact magnetic Lagrangians such that, when this unique measure is supported on a periodic orbit, this orbit is hyperbolic and its stable and unstable manifolds intersect transversally. This result is a version of an analogous theorem, for Tonelli Lagrangians, proven in [6].

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ACKNOWLEDGMENTS

I am grateful to Mário Jorge Dias Carneiro and José Antônio Gonçalves Miranda for several helpful comments and suggestions. I also thank to FAPEMIG-BRAZIL, which supported partially this work.

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Correspondence to Alexandre Rocha.

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MSC2010

37J50,70H09

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Rocha, A. Generic Properties of Mañé’s Set of Exact Magnetic Lagrangians. Regul. Chaot. Dyn. 26, 293–304 (2021). https://doi.org/10.1134/S1560354721030060

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