Integral representation for bracket-generating multi-flows

Abstract: If $f_1,f_2$ are smooth vector fields on an open subset of an Euclidean space and $[f_1,f_2]$ is their Lie bracket, the asymptotic formula $$\Psi_{[f_1,f_2]}(t_1,t_2)(x) - x =t_1t_2 f_1,f_2 +o(t_1t_2),$$ where we have set $ \Psi_{[f_1,f_2]}(t_1,t_2)(x) := \exp(-t_2f_2)\circ\exp(-t_1f_1)\circ\exp(t_2f_2)\circ\exp(t_1f_1)(x)$, is valid for all $t_1,t_2$ small enough. In fact, the integral, exact formula \begin{equation}\label{abstractform} \Psi_{[f_1,f_2]}(t_1,t_2)(x) - x = \int_0^{t_1}\int_0^{t_2}[f_1,f_2]^{(s_2,s_1)} (\Psi(t_1,s_2)(x))ds_1\,ds_2 , \end{equation} where $ [f_1,f_2]^{(s_2,s_1)}(y) := D\Big(\exp(s_1f_1)\circ \exp(s_2f_2{{)}}\Big)^{-1}\cdot f_1,f_2, $ with ${{y = \Psi(t_1,s_2)(x)}}$ has also been proven. Of course the integral formula can be regarded as an improvement of the asymptotic formula. In this paper we show that an integral representation holds true for any iterated bracket made from elements of a family of vector fields ${f_1,\dots,f_{{k}}}$. In perspective, these integral representations might lie at the basis for extensions of asymptotic formulas involving nonsmooth vector fields.