2019/10/10 by Enrico Le Donne, Donne, Enrico Le, Roger Züst +1 · 1 citation
Computer Science · Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Morphological variations and asymmetry #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1910.04589
openalex publication_date 2019/10/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Rn. 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 Rn 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.