Space of signatures as inverse limits of Carnot groups^*

¹ Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy.
² University of Jyväskylä, Department of Mathematics and Statistics, P.O. Box (MaD), 40014, Finland.
³ Mathematical Institute, University of Bern, Alpeneggstrasse 22, 3012 Bern, Switzerland.

^** Corresponding author: ledonne@msri.org

Received: 29 April 2020
Accepted: 5 April 2021

Abstract

We formalize the notion of limit of an inverse system of metric spaces with 1-Lipschitz projections having unbounded fibers. The construction is applied to the sequence of free Carnot groups of fixed rank n and increasing step. In this case, the limit space is in correspondence with the space of signatures of rectifiable paths in ℝⁿ, as introduced by Chen. Hambly-Lyons’s result on the uniqueness of signature implies that this space is a geodesic metric tree. As a particular consequence we deduce that every path in 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.