On the inner automorphisms of a singular foliation

A singular foliation in the sense of Androulidakis and Skandalis is an involutive and locally finitely generated module of compactly supported vector fields on a manifold. An automorphism of a singular foliation is a diffeomorphism that preserves the module. In this note, we give a proof of the (surprisingly non-trivial) fundamental fact that the time-one flow of an element of a singular foliation (i.e. its exponential) is an automorphism of the singular foliation. This fact was previously proven in Androulidakis and Skandalis (J Reine Angew Math 626:1–37, 2009 ) using an infinite dimensional argument (involving differential operators), and the purpose of this note is to complement that proof with a finite dimensional proof in which the problem is reduced to solving an elementary ODE.