作为卡诺群逆极限的签名空间
Space of signatures as inverse limits of Carnot groups
摘要：
本文给出了度量空间中具有无界纤维的$1$ -Lipschitz投影逆系统的极限概念。其目的是使用子黎曼群度量空间的签名的可求长路径在欧几里得空间，介绍了陈。构造性极限空间在具有1-Lipschitz映射的点度量空间范畴中具有泛性质。在一般的设置一些度量性质进行了讨论，如测地线和电梯的存在。次度量的概念将发挥至关重要的作用。该构造适用于固定秩$n$和递增步长的自由卡诺群序列。在这种情况下，这样的极限空间对应于$\mathbb R^n$中的可求长路径的签名空间。Hambly-Lyons关于签名唯一性的结果意味着这个空间是一个测地线度量树，在每个点上以无穷价分支。作为一个特殊的结果，我们推断，每一个路径在$\mathbb R^n$可以近似的投影的一些测地线在一些卡诺群的秩$n$，给出了一个证据，证明复杂的亚黎曼测地线增加的步骤。
英文摘要：
We formalize the notion of limit of an inverse system of metric spaces with $1$-Lipschitz projections having unbounded fibers. The purpose is to use sub-Riemannian groups for metrizing the space of signatures of rectifiable paths in Euclidean spaces, as introduced by Chen. The constructive limit space has the universal property in the category of pointed metric spaces with 1-Lipschitz maps. In the general setting some metric properties are discussed such as the existence of geodesics and lifts. The notion of submetry will play a crucial role. The construction is applied to the sequence of free Carnot groups of fixed rank $n$ and increasing step. In this case, such limit space is in correspondence with the space of signatures of rectifiable paths in $\mathbb R^n$. Hambly-Lyons's result on the uniqueness of signature implies that this space is a geodesic metric tree that brunches at every point with infinite valence. As a particular consequence we deduce that every path in $\mathbb R^n$ can be approximated by projections of some geodesics in some Carnot group of rank $n$, giving an evidence that the complexity of sub-Riemannian geodesics increases with the step.
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