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Functions whose set of critical points is an arc

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Abstract

Let \(M\) be a \(C^{\infty }\) connected closed manifold with \(\mathrm{dim }(M)\ge 2\). Using tools developed by Körner in (J Lond Math Soc (2) 38(3):442–452, 1988) we prove that the subset of functions \(f\) in \(C^1(M,\mathbb R )\) such that the set of critical points of \(f\) is an arc is dense in \(C^{0}(M,\mathbb R )\). We then present applications in dynamics.

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Acknowledgments

The idea of this work came from a discussion with R. Deville who introduced the author to the work of Körner. The author also wants to thank T.W. Körner for sending a paper version of [5] and A. Fathi for his fruitfull comments and suggestions. Finally, the author thanks F. Laudenbach and M.Mazzucchelli who contributed to the final form of this paper. This work was supported by the ANR KAM faible (Project BLANC07-3-187245, Hamilton-Jacobi and Weak KAM theory).

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  1. Lyon, France

    Pierre Pageault

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  1. Pierre Pageault
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Correspondence to Pierre Pageault.

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Pageault, P. Functions whose set of critical points is an arc. Math. Z. 275, 1121–1134 (2013). https://doi.org/10.1007/s00209-013-1173-6

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  • DOI: https://doi.org/10.1007/s00209-013-1173-6

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