Accessibility and foliations with singularities

Abstract

Recently, Sussmann has proved that the accessible sets of a system of vectorfields on a C 00 manifold M are immersed submanifolds of M [1], [2]. In this paper we state a general theorem on accessible sets of collections of 'arrows' and indicate how it implies (a) the above result; (b) the fact that the orbits of an arbitrary 'isotopically connected' subgroup of Diflf(M) form a foliation with singularities; and (c) a similar result for groupoids of germs of local diffeomorphisms. The complete proofs will be published elsewhere.