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

We consider a family ${mathcal{H}}:= {X_1, dots, X_m}$ of C ¹ vector fields in ? ⁿ and we discuss the associated ${mathcal{H}}$ -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 ${mathcal{H}}$ -orbit ${mathcal{O}}$ 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.