A dynamical approach to the Sard problem in Carnot groups

Abstract

We introduce a dynamical-systems approach for the study of the Sard problem in sub-Riemannian Carnot groups. We show that singular curves can be obtained by concatenating trajectories of suitable dynamical systems. As an applications, we positively answer the Sard problem in some classes of Carnot groups.

The Sard problem in step 2 and in filiform Carnot groups,

“…the set of critical values of the endpoint map, is negligible or not; we refer to Section 2 for precise definitions. Despite such a simple formulation, only very partial results are known [1,5,6,16,18] even in settings with a rich structure such as Carnot groups [3,7,9,10,11,12,15]. The goal of this note is to provide a contribution in two meaningful classes of Carnot groups: those with nilpotency step 2, and filiform ones.…”

Subdifferentials and minimizing Sard conjecture in sub-Riemannian geometry

“…(iii) The sub-Riemannian structure (∆, g) over M corresponds to a Carnot group of step ⩽ 3 (see [29]) or of rank 2 and step 4 (see [18]). The latter case verifies indeed the stronger Sard Conjecture.…”

“…The latter case verifies indeed the stronger Sard Conjecture. We refer the reader to [17,29] for a few other specific examples of Carnot groups satisfying the Sard Conjecture and to [3,16,14,15,17,18,29,32,34,37,38] for further details, results and discussions on that conjecture.…”

ODE trajectories as abnormal curves in Carnot groups

“…Recently, both the regularity and Sard problems have seen progress through the study of abnormal curves from a dynamical systems viewpoint. A class of potentially minimizing abnormal curves in rank 2 sub-Riemannian structures was proved to have at least C 1 -regularity [BCJ + 20], and the Sard Conjecture was proved in Carnot groups of rank 2 step 4 and rank 3 step 3 [BV20]. The idea common to both articles is that differentiating the identities defining abnormal curves leads to an ODE system that some reparametrization of the control of the abnormal curve will satisfy.…”

“…The idea common to both articles is that differentiating the identities defining abnormal curves leads to an ODE system that some reparametrization of the control of the abnormal curve will satisfy. Then using normal forms for the resulting ODEs, the authors of [BCJ + 20] and [BV20] were able to prove their respective claims by studying trajectories of finitely many explicit ODE systems.…”