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Feleqi, Ermal; Rampazzo, Franco (2017)
Publisher: American Institute of Mathematical Sciences
Languages: English
Types: Preprint
Subjects: QA, 34G20, 34A26, Mathematics - Dynamical Systems

Classified by OpenAIRE into

arxiv: Mathematics::Commutative Algebra, Mathematics::General Topology, High Energy Physics::Experiment
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 $$\Psi_{[f_1,f_2]}(t_1,t_2)(x) - x =t_1t_2 [f_1,f_2](x) +o(t_1t_2),$$ 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](\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 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.
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    • [1] A. A. Agraˇcev and R. V. Gamkrelidze, Exponential representation of flows and a chronological enumeration, Mat. Sb. (N.S.), 107 (1978), 467-532, 639.
    • [2] A. A. Agraˇcev and R. V. Gamkrelidze, Chronological algebras and nonstationary vector fields, in Problems in geometry, (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 11 (1980), 135-176, 243.
    • [3] M. Bramanti, L. Brandolini and M. Pedroni, Basic properties of nonsmooth H¨ormander's vector fields and Poincar´e's inequality, Forum Math., 25 (2013), 703-769.
    • [4] A. Montanari and D. Morbidelli, Nonsmooth Ho¨rmander vector fields and their control balls, Trans. Amer. Math. Soc., 364 (2012), 2339-2375.
    • [5] A. Montanari and D. Morbidelli, Almost exponential maps and integrability results for a class of horizontally regular vector fields, Potential Anal., 38 (2013), 611-633.
    • [6] A. Montanari and D. Morbidelli, Step-s involutive families of vector fields, their orbits and the Poincar´e inequality, J. Math. Pures Appl. (9), 99 (2013), 375-394.
    • [7] A. Montanari and D. Morbidelli, Generalized Jacobi identities and ball-box theorem for horizontally regular vector fields, J. Geom. Anal., 24 (2014), 687-720.
    • [8] F. Rampazzo and H. J. Sussmann, Set-valued differentials and a nonsmooth version of ChowRashevski's theorem, in Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, FL, December 2001, IEEE Publications, (2001), 2613-2618.
    • [9] F. Rampazzo and H. J. Sussmann, Commutators of flow maps of nonsmooth vector fields, J. Differential Equations, 232 (2007), 134-175.
    • [10] E. T. Sawyer and R. L. Wheeden, H¨older continuity of weak solutions to subelliptic equations with rough coefficients, Mem. Amer. Math. Soc., 180 (2006), x+157pp.
    • Received May 2014; revised September 2014.
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