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$$, as introduced by Chen. Hambly-Lyons’s result on the uniqueness of signature implies that this space is a geodesic metric tree. As a particular consequence we deduce that every path in ℝ$$ 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.
Accepté le :
Première publication :
Publié le :
Keywords: Signature of paths, inverse limit, path lifting property, submetry, metric tree, Carnot group, free nilpotent group, sub-Riemannian distance
@article{COCV_2021__27_1_A39_0,
author = {Le Donne, Enrico and Z\"ust, Roger},
title = {Space of signatures as inverse limits of {Carnot} groups},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021040},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021040/}
}
TY - JOUR AU - Le Donne, Enrico AU - Züst, Roger TI - Space of signatures as inverse limits of Carnot groups JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021040/ DO - 10.1051/cocv/2021040 LA - en ID - COCV_2021__27_1_A39_0 ER -
%0 Journal Article %A Le Donne, Enrico %A Züst, Roger %T Space of signatures as inverse limits of Carnot groups %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021040/ %R 10.1051/cocv/2021040 %G en %F COCV_2021__27_1_A39_0
Le Donne, Enrico; Züst, Roger. Space of signatures as inverse limits of Carnot groups. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 37. doi: 10.1051/cocv/2021040
[1] and , Currents in metric spaces. Acta Math. 185 (2000) 1–80.
[2] , , and , The signature of a rough path: Uniqueness. Adv. Math. 293 (2016) 720–737.
[3] , and , A course in metric geometry. Vol. 33 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2001).
[4] , Iterated integrals and exponential homomorphisms. Proc. Lond. Math. Soc. 4 (1954) 502–512.
[5] , Integration of paths – a faithful representation of paths by non-commutative formal power series. Trans. Am. Math. Soc. 89 (1958) 395–407.
[6] and , Realization of metric spaces as inverse limits, and bilipschitz embedding in L1. Geom. Funct. Anal. 23 (2013) 96–133.
[7] and , Multidimensional Stochastic Processes as Rough Paths. Vol. 120 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2010).
[8] and , Uniqueness for the signature of a path of bounded variation and the reduced path group. Ann. Math. 171 (2010) 109–167.
[9] , Über lineare Funktionaloperationen. Sitzsber. Ahad. Wiss. Wien 121 (1912) 265–297.
[10] , The master field on the plane. Astérisque 388 (2017) ix+201.
[11] and , A topological characterization of R-trees. Trans. Am. Math. Soc. 320 (1990) 395–415.
[12] , An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. 67 (1936) 251–282.
Cité par Sources :
E.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’).
