Space of signatures as inverse limits of Carnot groups

Le Donne, Enrico; Züst, Roger (2021). Space of signatures as inverse limits of Carnot groups. ESAIM: COCV, 27(37), p. 37. EDP Sciences 10.1051/cocv/2021040

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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 R^n, 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 R^n 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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