Infinitely many periodic orbits just above the Mañé critical value on the 2-sphere

Abstract

We introduce a new critical value $c_{\infty }(L)$ for Tonelli Lagrangians $L$ on the tangent bundle of the 2-sphere without minimizing measures supported on a point. We show that $c_{\infty }(L)$ is strictly larger than the Mañé critical value $c(L)$, and on every energy level $e\in (c(L),c_{\infty }(L))$ there exist infinitely many periodic orbits of the Lagrangian system of $L$, one of which is a local minimizer of the free-period action functional. This has applications to Finsler metrics of Randers type on the 2-sphere. We show that, under a suitable criticality assumption on a given Randers metric, after rescaling its magnetic part with a sufficiently large multiplicative constant, the new metric admits infinitely many closed geodesics, one of which is a waist. Examples of critical Randers metrics include the celebrated Katok metric.

The results in this preprint are now incorporated in our work arXiv:1705.02488 with Luca Asselle. This preprint will not be submitted for publication

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