On finite quotient Aubry set for generic geodesic flows

TL;DR

This paper proves that for a generic set of conformal metrics on a closed manifold, the associated geodesic flows have finitely many ergodic minimizing measures, leading to a finite quotient Aubry set for each cohomology class.

Contribution

It establishes the finiteness of ergodic c-minimizing measures and quotient Aubry sets for generic conformal metrics in geodesic flows.

Findings

Existence of a residual set of conformal metrics with finite ergodic measures.

Finite quotient Aubry sets for generic geodesic flows.

Finiteness holds for all cohomology classes.

Abstract

We study the structure of the Mather and Aubry sets for the family of lagrangians given by the kinetic energy associated to a riemannian metric $ g$ on a closed manifold $ M$. In this case the Euler-Lagrange flow is the geodesic flow of $(M,g)$. We prove that there exists a residual subset $ \mathcal G$ of the set of all conformal metrics to $g$, such that, if $ \overline g \in \mathcal G$ then the corresponding geodesic flow has a finitely many ergodic c-minimizing measures, for each non-trivial cohomology class $ c \in H^1(M,\mathbb{R})$. This implies that, for any $ c \in H^1(M,\mathbb{R})$, the quotient Aubry set for the cohomology class c has a finite number of elements for this particular family of lagrangian systems.