The Sard conjecture on Martinet surfaces

Third order open mapping theorems and applications to the end-point map

“…Φ (15) = F ′ [φ (15) ] + c 69 F ′′ [φ (6) , φ (9) ] + c 78 F ′′ [φ (7) , φ (8) ] + O(ε) Φ (16) = F ′ [φ (16) ] + c 6,10 F ′′ [φ (6) , φ (10) ] + c 79 F ′′ [φ (7) , φ (9) ] + c 88 F ′′ [φ (8) , φ (8) ] + O(ε) Φ (17) = F ′ [φ (17) ] + c 6,11 F ′′ [φ (6) , φ (11) ] + c 7,10 F ′′ [φ (7) , φ (10) ] + c 89 F ′′ [φ (8) , φ (9) ] + O(ε).…”

“…The Sard's problem investigates the size (dimension, measure, structure) of the set of points of M that are reachable from q 0 by singular curves. Even though Sard's theorem does not hold in infinite-dimensional spaces [19], it is expected that for the end-point map this set is not too big, see [22,10,32].…”

Measure contraction properties for two-step analytic sub-Riemannian structures and Lipschitz Carnot groups

“…Let us now assume that (2.2) is satisfied. By definition of div µ Z, for every x ∈ M and any measurable set A ⊂ O x , we have for every t ≥ 0 (see for example, see [12…”

Strong Sard conjecture and regularity of singular minimizing geodesics for analytic sub-Riemannian structures in dimension 3

“…Hence, it is natural to investigate what one can say without a genericity assumption, both for the Sard Conjecture and for its Strong version. Recently, in [9] the first and fourth authors proved that the Sard Conjecture in dimension 3 holds whenever at least one of these two conditions holds: either is smooth; or the distribution satisfies some suitable tangency assumption over the set of singularities of . Unfortunately, the approach used in [9] strongly relies on the assumptions listed above and cannot be adapted to study the general case, nor to attack the Strong Sard Conjecture.…”