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 of the set of all conformal metrics to g, such that, if g ¯ ∈ G $ \overline g \in G$ then the corresponding geodesic flow has finitely many ergodic c-minimizing measures, for each non-zero cohomology class c ∈ H 1 ( M , ℝ ) $ c \in H^{1}(M,\mathbb {R})$ . This implies that, for any c ∈ H 1 ( M , ℝ ) $ 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.

中文翻译：

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通用测地线流的有限商 Aubry 集

我们研究了由与闭合流形 M 上的黎曼度量 g 相关联的动能给出的拉格朗日族的 Mather 和 Aubry 集的结构。在这种情况下，欧拉-拉格朗日流是 ( M , g ) 的测地线流。我们证明存在所有对 g 的共形度量的残差子集，这样，如果 g¯ ∈ G $ \overline g \in G$ 那么相应的测地线流具有有限多个遍历 c-最小化度量，对于每个非零上同调类 c ∈ H 1 ( M , ℝ ) $ c \in H^{1}(M,\mathbb {R})$ 。这意味着，对于任何 c ∈ H 1 ( M , ℝ ) $ c \in H^{1}(M,\mathbb {R})$ ，上同调类 c 的商 Aubry 集具有有限数量的元素对于这个特殊的拉格朗日系统家族。