论文标题
Weyl Law的对数改善和封闭的大地测量数量的指数界限是主要的
Logarithmic improvements in the Weyl law and exponential bounds on the number of closed geodesics are predominant
论文作者
论文摘要
令$ m $为无边界的尺寸$ d $的平滑紧凑型歧管。我们介绍了Riemannian指标在$ M $上的占主导地位的概念，这是一个类似于完整的Lebesgue度量的概念，尤其意味着密度。我们表明，对于主要的度量，小于$ t $的长度的闭合大地测量学的数量具有$ t $的伸展指数上限。 此外，我们研究了Weyl定律的主要指标。 Weyl Law指出，Laplace-Beltrami特征值小于$λ^2 $渐近至$cλ^d $，带有$ o（λ^{d-1}）$ error。我们表明，对于一个主要的度量，对错误的估计值可以通过$ \logλ$的功率提高。在作者在Weyl定律的情况下采用了最新结果后，这些估计得出了对主要指标集的几乎封闭轨道的非分类特性的研究。
Let $M$ be a smooth compact manifold of dimension $d$ without boundary. We introduce the concept of predominance for Riemannian metrics on $M$, a notion analogous to full Lebesgue measure which, in particular, implies density. We show that for a predominant metric, the number of closed geodesics of length smaller than $T$ has a stretched exponential upper bound in $T$. In addition, we study remainders in the Weyl law for predominant metrics. The Weyl law states that the number of Laplace-Beltrami eigenvalues smaller than $λ^2$ is asymptotic to $Cλ^d$ with an $O(λ^{d-1})$ error. We show that, for a predominant metric, the estimate on the error can by improved by a power of $\log λ$. After an application of recent results of the authors in the case of the Weyl law, these estimates follow from a study of the non-degeneracy properties of nearly closed orbits for predominant sets of metrics.
