Search results for "inverse limit"

showing 2 items of 2 documents

Space of signatures as inverse limits of Carnot groups

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.…

Carnot groupsignature of pathsryhmäteoriametric treeinverse limitsub-Riemannian distancedifferentiaaligeometria510 Mathematicspath lifting propertysubmetryMathematics::Metric GeometryMathematics::Differential Geometrymittateoriafree nilpotent groupstokastiset prosessit
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Internal inverse limits and retractions

2015

We establish equivalences between compacta that admit a sequence of retractions that converge uniformly to the identity map and compacta that are inverse limits on subcompacta with retractions for bonding maps. We give partial answers to questions of Charatonik and Prajs, and of Krasinkiewicz. Our results are related to and use results from another paper of the authors \cite{mp}.

Discrete mathematicsSequenceGeneral Mathematics54A20Inverse$r$-maps54F6554C15retractions54F15CalculusIdentity functionInternal inverse limitMathematics
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