Infinitely many periodic orbits just above the Ma\~n\'e critical value on the 2-sphere

TL;DR

This paper introduces a new critical value for Tonelli Lagrangians on the 2-sphere, proving the existence of infinitely many periodic orbits between two critical levels, with applications to Randers Finsler metrics.

Contribution

It defines a new critical value $c_ ty(L)$, shows it exceeds the Mañé critical value, and proves the existence of infinitely many periodic orbits in this energy range, including a local minimizer.

Findings

Existence of infinitely many periodic orbits on energy levels between $c(L)$ and $c_ ty(L)$.

Introduction of a new critical value $c_ ty(L)$ exceeding the Mañé critical value.

Application to Randers metrics, including the Katok metric, showing infinitely many closed geodesics.

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 Ma\~n\'e 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.