Integral representations for bracket-generating multi-flows

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

Integral representations for bracket-generating multi-flows

Ermal Feleqi ^1, and Franco Rampazzo ^1,

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

Abstract
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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 \begin{equation}\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) \end{equation} 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 \begin{equation}\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) \end{equation} 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.

Keywords: asymptotic formulas, Chow's theorem, integral formulas, multi-flows, Iterated Lie brackets, low smoothness hypotheses.

Mathematics Subject Classification: Primary: 34A26, 34H05; Secondary: 93B0.

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.

show all references

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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