INTEGRAL REPRESENTATIONS FOR BRACKET-GENERATING MULTI-FLOWS

摘要

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 Ψ_([f_1,f_2])(t_1, t_2)(x) -x = t_1t_2[f_1,f_2](x) + o(t_1t_2),t(1) where we have set Ψ_([f_1,f_2])(t_1,t_2)(x)def/= exp(-t_2f_2) o exp(-t_1 f_1) o exp(t_2f_2) o exp(t_1f_1)(x), is valid for all t_1,t_2 small enough. In fact, the integral, exact formula Ψ_([f_1,f_2])(t_1.t_2)(x)-x= ∫_0~(t_1)∫_0~(t_2)[f_1,f_2]~(s_2,s_1)(Ψ(t_1,s_2)(x))ds_1ds_2, (2) where [f_1,f_2]~(s_2,s_1) (y)def/=D(exp(s_1f_1) oexp(S_2f_2)))~(-1) (y) • [f_1,f_2](exp(s_1f_1) o exp(s_2f_2)(y)), has also been proven. Of course (2) can be regarded as an improvement of (1). In this paper we show that an integral representation like (2) holds true for any iterated Lie bracket made of elements of a family f_1, • • •, f_m of vector fields. In perspective, these integral representations might lie at the basis for extensions of asymptotic formulas involving non-smooth vector fields.