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Waist theorems for Tonelli systems in higher dimensions

Abstract : We study the periodic orbits problem on energy levels of Tonelli Lagrangian systems over configuration spaces of arbitrary dimension. We show that, when the fundamental group is finite and the Lagrangian has no stationary orbit at the Ma\~n\'e critical energy level, there is a waist on every energy level just above the Mañé critical value. With a suitable perturbation with a potential, we show that there are infinitely many periodic orbits on every energy level just above the Mañé critical value, and on almost every energy level just below. Finally, we prove the Tonelli analogue of a closed geodesics result due to Ballmann-Thorbergsson-Ziller.
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https://hal-ens-lyon.archives-ouvertes.fr/ensl-02352679
Contributor : Marco Mazzucchelli <>
Submitted on : Thursday, November 7, 2019 - 12:59:48 AM
Last modification on : Friday, August 7, 2020 - 4:00:07 PM

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Luca Asselle, Marco Mazzucchelli. Waist theorems for Tonelli systems in higher dimensions. manuscripta mathematica, Springer Verlag, 2020, 163 (1-2), pp.185-199. ⟨10.1007/s00229-019-01154-5⟩. ⟨ensl-02352679⟩

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