6533b7dafe1ef96bd126f6f2
RESEARCH PRODUCT
Space of signatures as inverse limits of Carnot groups
Enrico Le DonneRoger Z��stsubject
Carnot groupsignature of pathsryhmäteoriametric treeinverse limitsub-Riemannian distancedifferentiaaligeometria510 Mathematicspath lifting propertysubmetryMathematics::Metric GeometryMathematics::Differential Geometrymittateoriafree nilpotent groupstokastiset prosessitdescription
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. peerReviewed
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2021-01-01 |