Volume 27, 2021
|Number of page(s)||14|
|Published online||30 April 2021|
Space of signatures as inverse limits of Carnot groups*
Dipartimento di Matematica, Università di Pisa,
Largo B. Pontecorvo 5,
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: firstname.lastname@example.org
Accepted: 5 April 2021
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.
Mathematics Subject Classification: 22E25 / 53C17 / 49Q15 / 28A75 / 60H05
Key words: Signature of paths / inverse limit / path lifting property / submetry / metric tree / Carnot group / free nilpotent group / sub-Riemannian distance
© EDP Sciences, SMAI 2021
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