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 \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.
Citation: Ermal Feleqi, Franco Rampazzo. Integral representations for bracket-generating multi-flows. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4345-4366. doi: 10.3934/dcds.2015.35.4345
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show all references

References:
[1]

Mat. Sb. (N.S.), 107 (1978), 467-532, 639.  Google Scholar

[2]

in Problems in geometry, (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 11 (1980), 135-176, 243.  Google Scholar

[3]

Forum Math., 25 (2013), 703-769. doi: 10.1515/form.2011.133.  Google Scholar

[4]

Trans. Amer. Math. Soc., 364 (2012), 2339-2375. doi: 10.1090/S0002-9947-2011-05395-X.  Google Scholar

[5]

Potential Anal., 38 (2013), 611-633. doi: 10.1007/s11118-012-9289-6.  Google Scholar

[6]

J. Math. Pures Appl. (9), 99 (2013), 375-394. doi: 10.1016/j.matpur.2012.09.005.  Google Scholar

[7]

J. Geom. Anal., 24 (2014), 687-720. doi: 10.1007/s12220-012-9351-z.  Google Scholar

[8]

in Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, FL, December 2001, IEEE Publications, (2001), 2613-2618. Google Scholar

[9]

J. Differential Equations, 232 (2007), 134-175. doi: 10.1016/j.jde.2006.04.016.  Google Scholar

[10]

Mem. Amer. Math. Soc., 180 (2006), x+157pp. doi: 10.1090/memo/0847.  Google Scholar

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