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 Mañé 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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Notes
In [14], the authors call “weakly monotonous quasi elliptic” a periodic orbit of twist type.
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Acknowledgements
Marco Mazzucchelli is grateful to Alessio Figalli and Ludovic Rifford for pointing out the argument in [12, page 935] in order to obtain the hyperbolicity of the periodic orbit in their result [20, Theorem 1.2]. Both authors are grateful to the anonymous referee for her/his careful reading of the paper, and for spotting a few inaccuracies in the first draft. Luca Asselle is partially supported by the DFG-Grants AB 360/2-1 “Periodic orbits of conservative systems below the Mañé critical energy value” and AS 546/1-1 “Morse theoretical methods in Hamiltonian dynamics”.
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Asselle, L., Mazzucchelli, M. Waist theorems for Tonelli systems in higher dimensions. manuscripta math. 163, 185–199 (2020). https://doi.org/10.1007/s00229-019-01154-5
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DOI: https://doi.org/10.1007/s00229-019-01154-5