# American Institute of Mathematical Sciences

September  2015, 35(9): 4345-4366. doi: 10.3934/dcds.2015.35.4345

## Integral representations for bracket-generating multi-flows

 1 Dipartimento di Matematica, Università degli Studi di Padova, Via Trieste 63 - 35121 - Padova (PD), Italy, Italy

Received  May 2014 Revised  September 2014 Published  April 2015

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 $$\label{abstract:EQ} \Psi_{[f_1,f_2]}(t_1,t_2)(x) - x =t_1t_2 [f_1,f_2](x) +o(t_1t_2), \, (1)$$ where we have set $\Psi_{[f_1,f_2]}(t_1,t_2)(x) \overset{\underset{\mathrm{def}}{}}{=} \exp(-t_2 f_2)\circ \exp(-t_1f_1) \circ \exp(t_2f_2) \circ \exp(t_1f_1)(x)$, is valid for all $t_1,t_2$ small enough. In fact, the integral, exact formula $$\label{abstract:EQ} \Psi_{[f_1,f_2]}(t_1,t_2)(x) - x = \int_0^{t_1}\int_0^{t_2}[f_1,f_2]^{(s_2,s_1)} (\Psi(t_1,s_2)(x))ds_1\,ds_2 , (2)$$ where $[f_1,f_2]^{(s_2,s_1)}(y) \overset{\underset{\mathrm{def}}{}}{=} D (\exp(s_1f_1) \circ \exp(s_2f_2)))^{-1}(y) \cdot [f_1,f_2](\exp (s_1f_1) \circ \exp(s_2f_2)(y) ),$ 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 ${f_1,\dots,f_m}$ of vector fields. In perspective, these integral representations might lie at the basis for extensions of asymptotic formulas involving non-smooth vector fields.
Citation: Ermal Feleqi, Franco Rampazzo. Integral representations for bracket-generating multi-flows. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4345-4366. doi: 10.3934/dcds.2015.35.4345
##### References:
 [1] A. A. Agračev and R. V. Gamkrelidze, Exponential representation of flows and a chronological enumeration, Mat. Sb. (N.S.), 107 (1978), 467-532, 639. [2] A. A. Agračev 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 Hörmander's vector fields and Poincaré's inequality, Forum Math., 25 (2013), 703-769. doi: 10.1515/form.2011.133. [4] A. Montanari and D. Morbidelli, Nonsmooth Hörmander vector fields and their control balls, Trans. Amer. Math. Soc., 364 (2012), 2339-2375. doi: 10.1090/S0002-9947-2011-05395-X. [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. doi: 10.1007/s11118-012-9289-6. [6] A. Montanari and D. Morbidelli, Step-$s$ involutive families of vector fields, their orbits and the Poincaré inequality, J. Math. Pures Appl. (9), 99 (2013), 375-394. doi: 10.1016/j.matpur.2012.09.005. [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. doi: 10.1007/s12220-012-9351-z. [8] F. Rampazzo and H. J. Sussmann, Set-valued differentials and a nonsmooth version of Chow-Rashevski'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. doi: 10.1016/j.jde.2006.04.016. [10] E. T. Sawyer and R. L. Wheeden, Hölder continuity of weak solutions to subelliptic equations with rough coefficients, Mem. Amer. Math. Soc., 180 (2006), x+157pp. doi: 10.1090/memo/0847.

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##### References:
 [1] A. A. Agračev and R. V. Gamkrelidze, Exponential representation of flows and a chronological enumeration, Mat. Sb. (N.S.), 107 (1978), 467-532, 639. [2] A. A. Agračev 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 Hörmander's vector fields and Poincaré's inequality, Forum Math., 25 (2013), 703-769. doi: 10.1515/form.2011.133. [4] A. Montanari and D. Morbidelli, Nonsmooth Hörmander vector fields and their control balls, Trans. Amer. Math. Soc., 364 (2012), 2339-2375. doi: 10.1090/S0002-9947-2011-05395-X. [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. doi: 10.1007/s11118-012-9289-6. [6] A. Montanari and D. Morbidelli, Step-$s$ involutive families of vector fields, their orbits and the Poincaré inequality, J. Math. Pures Appl. (9), 99 (2013), 375-394. doi: 10.1016/j.matpur.2012.09.005. [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. doi: 10.1007/s12220-012-9351-z. [8] F. Rampazzo and H. J. Sussmann, Set-valued differentials and a nonsmooth version of Chow-Rashevski'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. doi: 10.1016/j.jde.2006.04.016. [10] E. T. Sawyer and R. L. Wheeden, Hölder continuity of weak solutions to subelliptic equations with rough coefficients, Mem. Amer. Math. Soc., 180 (2006), x+157pp. doi: 10.1090/memo/0847.
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