Space of signatures as inverse limits of Carnot groups
Le Donne, E., & Züst, R. (2021). Space of signatures as inverse limits of Carnot groups. ESAIM : Control, Optimisation and Calculus of Variations, 27, Article 37. https://doi.org/10.1051/cocv/2021040
© EDP Sciences, SMAI 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.
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Related funder(s)Academy of Finland; European Commission
Funding program(s)Research post as Academy Research Fellow, AoF; Academy Project, AoF
The content of the publication reflects only the author’s view. The funder is not responsible for any use that may be made of the information it contains.
Additional information about fundingE.L.D. was partially supported by the Academy of Finland (grant 288501 ‘Geometry of subRiemannian groups’ and by grant 322898 ‘Sub-Riemannian Geometry via Metric-geometry and Lie-group Theory’) and by the European Research Council (ERC Starting Grant 713998 GeoMeG ‘Geometry of Metric Groups’).
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