Almost Exponential Maps and Integrability Results for a Class of Horizontally Regular Vector Fields

We consider a family of C (1) vector fields in a"e (n) and we discuss the associated -orbits. Namely, we assume that our vector fields belong to a horizontal regularity class and we require that a suitable s-involutivity assumption holds. Then we show that any -orbit is a C (1) immersed submanifold and it is an integral submanifold of the distribution generated by the family of all commutators up to length s. Our main tool is a class of almost exponential maps of which we discuss carefully some precise first order expansions.