# 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 & Continuous Dynamical Systems - A, 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.   Google Scholar [2] A. A. Agračev and R. V. Gamkrelidze, Chronological algebras and nonstationary vector fields,, in Problems in geometry, 11 (1980), 135.   Google Scholar [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.  doi: 10.1515/form.2011.133.  Google Scholar [4] A. Montanari and D. Morbidelli, Nonsmooth Hörmander vector fields and their control balls,, Trans. Amer. Math. Soc., 364 (2012), 2339.  doi: 10.1090/S0002-9947-2011-05395-X.  Google Scholar [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.  doi: 10.1007/s11118-012-9289-6.  Google Scholar [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.  doi: 10.1016/j.matpur.2012.09.005.  Google Scholar [7] A. Montanari and D. Morbidelli, Generalized Jacobi identities and ball-box theorem for horizontally regular vector fields,, J. Geom. Anal., 24 (2014), 687.  doi: 10.1007/s12220-012-9351-z.  Google Scholar [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, (2001), 2613.   Google Scholar [9] F. Rampazzo and H. J. Sussmann, Commutators of flow maps of nonsmooth vector fields,, J. Differential Equations, 232 (2007), 134.  doi: 10.1016/j.jde.2006.04.016.  Google Scholar [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).  doi: 10.1090/memo/0847.  Google Scholar

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.   Google Scholar [2] A. A. Agračev and R. V. Gamkrelidze, Chronological algebras and nonstationary vector fields,, in Problems in geometry, 11 (1980), 135.   Google Scholar [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.  doi: 10.1515/form.2011.133.  Google Scholar [4] A. Montanari and D. Morbidelli, Nonsmooth Hörmander vector fields and their control balls,, Trans. Amer. Math. Soc., 364 (2012), 2339.  doi: 10.1090/S0002-9947-2011-05395-X.  Google Scholar [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.  doi: 10.1007/s11118-012-9289-6.  Google Scholar [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.  doi: 10.1016/j.matpur.2012.09.005.  Google Scholar [7] A. Montanari and D. Morbidelli, Generalized Jacobi identities and ball-box theorem for horizontally regular vector fields,, J. Geom. Anal., 24 (2014), 687.  doi: 10.1007/s12220-012-9351-z.  Google Scholar [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, (2001), 2613.   Google Scholar [9] F. Rampazzo and H. J. Sussmann, Commutators of flow maps of nonsmooth vector fields,, J. Differential Equations, 232 (2007), 134.  doi: 10.1016/j.jde.2006.04.016.  Google Scholar [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).  doi: 10.1090/memo/0847.  Google Scholar
 [1] Linh V. Nguyen. A family of inversion formulas in thermoacoustic tomography. Inverse Problems & Imaging, 2009, 3 (4) : 649-675. doi: 10.3934/ipi.2009.3.649 [2] Roderick S. C. Wong, H. Y. Zhang. On the connection formulas of the third Painlevé transcendent. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 541-560. doi: 10.3934/dcds.2009.23.541 [3] Jérôme Rousseau, Paulo Varandas, Yun Zhao. Entropy formulas for dynamical systems with mistakes. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4391-4407. doi: 10.3934/dcds.2012.32.4391 [4] J. C. Alvarez Paiva and E. Fernandes. Crofton formulas in projective Finsler spaces. Electronic Research Announcements, 1998, 4: 91-100. [5] Matthew B. Rudd. Statistical exponential formulas for homogeneous diffusion. Communications on Pure & Applied Analysis, 2015, 14 (1) : 269-284. doi: 10.3934/cpaa.2015.14.269 [6] Dmitry Kleinbock, Barak Weiss. Dirichlet's theorem on diophantine approximation and homogeneous flows. Journal of Modern Dynamics, 2008, 2 (1) : 43-62. doi: 10.3934/jmd.2008.2.43 [7] Zvi Drezner, Carlton Scott. Approximate and exact formulas for the $(Q,r)$ inventory model. Journal of Industrial & Management Optimization, 2015, 11 (1) : 135-144. doi: 10.3934/jimo.2015.11.135 [8] Janusz Mierczyński, Wenxian Shen. Formulas for generalized principal Lyapunov exponent for parabolic PDEs. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1189-1199. doi: 10.3934/dcdss.2016048 [9] Paul Loya and Jinsung Park. On gluing formulas for the spectral invariants of Dirac type operators. Electronic Research Announcements, 2005, 11: 1-11. [10] Cuilian You, Le Bo. Option pricing formulas for generalized fuzzy stock model. Journal of Industrial & Management Optimization, 2020, 16 (1) : 387-396. doi: 10.3934/jimo.2018158 [11] Francis N. Castro, Carlos Corrada-Bravo, Natalia Pacheco-Tallaj, Ivelisse Rubio. Explicit formulas for monomial involutions over finite fields. Advances in Mathematics of Communications, 2017, 11 (2) : 301-306. doi: 10.3934/amc.2017022 [12] João Paulo da Silva, Julio López, Ricardo Dahab. Isogeny formulas for Jacobi intersection and twisted hessian curves. Advances in Mathematics of Communications, 2020, 14 (3) : 507-523. doi: 10.3934/amc.2020048 [13] Matilde Martínez, Shigenori Matsumoto, Alberto Verjovsky. Horocycle flows for laminations by hyperbolic Riemann surfaces and Hedlund's theorem. Journal of Modern Dynamics, 2016, 10: 113-134. doi: 10.3934/jmd.2016.10.113 [14] Giulia Cavagnari, Antonio Marigonda. Measure-theoretic Lie brackets for nonsmooth vector fields. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 845-864. doi: 10.3934/dcdss.2018052 [15] Michel L. Lapidus, Goran Radunović, Darko Žubrinić. Fractal tube formulas and a Minkowski measurability criterion for compact subsets of Euclidean spaces. Discrete & Continuous Dynamical Systems - S, 2019, 12 (1) : 105-117. doi: 10.3934/dcdss.2019007 [16] Tohru Wakasa, Shoji Yotsutani. Representation formulas for some 1-dimensional linearized eigenvalue problems. Communications on Pure & Applied Analysis, 2008, 7 (4) : 745-763. doi: 10.3934/cpaa.2008.7.745 [17] Marianito R. Rodrigo, Rogemar S. Mamon. Bond pricing formulas for Markov-modulated affine term structure models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020089 [18] Stefan Erickson, Michael J. Jacobson, Jr., Andreas Stein. Explicit formulas for real hyperelliptic curves of genus 2 in affine representation. Advances in Mathematics of Communications, 2011, 5 (4) : 623-666. doi: 10.3934/amc.2011.5.623 [19] Tatsuki Mori, Kousuke Kuto, Tohru Tsujikawa, Shoji Yotsutani. Representation formulas of solutions and bifurcation sheets to a nonlocal Allen-Cahn equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (8) : 4907-4925. doi: 10.3934/dcds.2020205 [20] Z. B. Ibrahim, N. A. A. Mohd Nasir, K. I. Othman, N. Zainuddin. Adaptive order of block backward differentiation formulas for stiff ODEs. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 95-106. doi: 10.3934/naco.2017006

2019 Impact Factor: 1.338