Integral representation for bracket-generating multi-flows

摘要

If $f_1,f_2$ are smooth vector fields on an open subset of an Euclidean spaceand $[f_1,f_2]$ is their Lie bracket, the asymptotic formula$$\Psi_{[f_1,f_2]}(t_1,t_2)(x) - x =t_1t_2 [f_1,f_2](x) +o(t_1t_2),$$ where wehave 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 validfor 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](\exp(s_1f_1)\circ \exp(s_2f_2){(y)}),$ with ${{y = \Psi(t_1,s_2)(x)}}$ has also been proven. Of course the integralformula can be regarded as an improvement of the asymptotic formula. In thispaper we show that an integral representation holds true for any iteratedbracket made from elements of a family of vector fields ${f_1,\dots,f_{{k}}}$.In perspective, these integral representations might lie at the basis forextensions of asymptotic formulas involving nonsmooth vector fields.