Resonance for loop homology of spheres

TL;DR

This paper establishes the existence of a well-defined global mean frequency for spheres of dimension greater than two, linking it to the distribution of closed geodesics via homology and resonance phenomena.

Contribution

It proves the existence of the limit of critical levels normalized by degree for homology classes on loop spaces of spheres, revealing resonance properties of closed geodesics.

Findings

Existence of the global mean frequency as deg(X) approaches infinity.

Either all high-degree homology classes are associated with geodesics of mean frequency equal to the global mean frequency, or infinitely many geodesics have mean frequencies converging to it.

Application of the Chas-Sullivan product and Goresky-Hingston results to prove resonance phenomena.

Abstract

A Riemannian or Finsler metric on a compact manifold M gives rise to a length function on the free loop space \Lambda M, whose critical points are the closed geodesics in the given metric. If X is a homology class on \Lambda M, the minimax critical level cr(X) is a critical value. Let M be a sphere of dimension >2, and fix a metric g and a coefficient field G. We prove that the limit as deg(X) goes to infinity of cr(X)/deg(X) exists. We call this limit the "global mean frequency" of M. As a consequence we derive resonance statements for closed geodesics on spheres; in particular either all homology on \Lambda M of sufficiently high degreee lies hanging on closed geodesics whose mean frequency (average index / length) equals the global mean frequency, or there is a sequence of infinitely many closed geodesics whose mean frequencies converge to the global mean frequency. The proof uses the Chas-Sullivan product and results of Goresky-Hingston [GH].