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July  2011, 29(3): 1031-1039. doi: 10.3934/dcds.2011.29.1031

Linearization of cohomology-free vector fields

 1 UFR de Mathématiques, Université de Lille 1 (USTL), F59655 Villeneuve d'Asq Cedex 2 Centro de Matemática, Facultad de Ciencias, Iguá 4225, 11400 Montevideo, Uruguay

Received  January 2010 Revised  July 2010 Published  November 2010

We study the cohomological equation for a smooth vector field on a compact manifold. We show that if the vector field is cohomology free, then it can be embedded continuously in a linear flow on an Abelian group.
Citation: Livio Flaminio, Miguel Paternain. Linearization of cohomology-free vector fields. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1031-1039. doi: 10.3934/dcds.2011.29.1031
References:
 [1] A. Avila and A. Kocsard, Cohomological equations and invariant distributions for minimal circle diffeomorphisms,, \arXiv{1002.3392}, (2010). Google Scholar [2] J. C. Baez and S. Sawin, Functional integration on spaces of connections,, J. Funct. Anal., 150 (1997), 1. doi: 10.1006/jfan.1997.3108. Google Scholar [3] K. T. Chen, Iterated path integrals,, Bull. Amer. Math. Soc., 83 (1977), 831. doi: 10.1090/S0002-9904-1977-14320-6. Google Scholar [4] W. Chen and M. Y. Chi, Hypoelliptic vector fields and almost periodic motions on the torus $T^n$,, Comm. Partial Differential Equations, 25 (2000), 337. doi: 10.1080/03605300008821516. Google Scholar [5] D. Damjanović, Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions,, J. Mod. Dyn., 1 (2007), 665. Google Scholar [6] D. Damjanović and A. Katok, Local rigidity of actions of higher rank abelian groups and KAM method,, Electron. Res. Announc. Amer. Math. Soc., 10 (2004), 142. doi: 10.1090/S1079-6762-04-00139-8. Google Scholar [7] D. Damjanović and A. Katok, Periodic cycle functionals and cocycle rigidity for certain partially hyperbolic $\bb R^k$ actions,, Discrete Contin. Dyn. Syst., 13 (2005), 985. doi: 10.3934/dcds.2005.13.985. Google Scholar [8] D. Damjanović and A. Katok, Local rigidity of restrictions of Weyl chamber flows,, C. R. Math. Acad. Sci. Paris, 344 (2007), 503. Google Scholar [9] R. de la Llave, J. M. Marco and R. Moriyón, Canonical perturbation theory of Anosov systems, and regularity results for the Livsic cohomology equation,, Bull. Amer. Math. Soc. (N.S.), 12 (1985), 91. Google Scholar [10] L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows,, Duke Math. J., 119 (2003), 465. doi: 10.1215/S0012-7094-03-11932-8. Google Scholar [11] L. Flaminio and G. Forni, On the cohomological equation for nilflows,, J. Mod. Dyn., 1 (2007), 37. Google Scholar [12] G. Forni, On the Greenfield-Wallach and Katok conjectures in dimension three,, in, 469 (2008), 197. Google Scholar [13] S. J. Greenfield and N. R. Wallach, Globally hypoelliptic vector fields,, Topology, 12 (1973), 247. doi: 10.1016/0040-9383(73)90011-6. Google Scholar [14] V. Guillemin and D. Kazhdan, On the cohomology of certain dynamical systems,, Topology, 19 (1980), 291. doi: 10.1016/0040-9383(80)90014-2. Google Scholar [15] V. Guillemin and D. Kazhdan, Some inverse spectral results for negatively curved $n$-manifolds,, in, XXXVI (1980), 153. Google Scholar [16] M. R. Herman, $L^2$ regularity of measurable solutions of a finite-difference equation of the circle,, Ergodic Theory Dynam. Systems, 24 (2004), 1277. doi: 10.1017/S0143385704000409. Google Scholar [17] S. Hurder, Problems on rigidity of group actions and cocycles,, Ergodic Theory Dynam. Systems, 5 (1985), 473. doi: 10.1017/S0143385700003084. Google Scholar [18] P. Iglesias, "Diffeology,", manuscript., (). Google Scholar [19] A. Katok, Cocycles, cohomology and combinatorial constructions in ergodic theory,, in, 69 (2001), 107. Google Scholar [20] A. Katok, Combinatorial constructions in ergodic theory and dynamics,, University Lecture Series, 30 (2003). Google Scholar [21] A. Kocsard, Cohomologically rigid vector fields: The Katok conjecture in dimension 3,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1165. doi: 10.1016/j.anihpc.2008.07.005. Google Scholar [22] R. U. Luz and N. M. dos Santos, Cohomology-free diffeomorphisms of low-dimension tori,, Ergodic Theory Dynam. Systems, 18 (1998), 985. doi: 10.1017/S0143385798108222. Google Scholar [23] S. Matsumoto, The parameter rigid flows on 3-manifolds,, in, 498 (2009), 135. Google Scholar [24] D. Mieczkowski, The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory,, J. Mod. Dyn., 1 (2007), 61. Google Scholar [25] F. R. Hertz and J. R. Hertz, Cohomology free systems and the first Betti number,, Discrete Contin. Dyn. Syst., 15 (2006), 193. doi: 10.3934/dcds.2006.15.193. Google Scholar [26] C. H. Taubes, The Seiberg-Witten equations and the Weinstein conjecture,, Geom. Topol., 11 (2007), 2117. doi: 10.2140/gt.2007.11.2117. Google Scholar

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References:
 [1] A. Avila and A. Kocsard, Cohomological equations and invariant distributions for minimal circle diffeomorphisms,, \arXiv{1002.3392}, (2010). Google Scholar [2] J. C. Baez and S. Sawin, Functional integration on spaces of connections,, J. Funct. Anal., 150 (1997), 1. doi: 10.1006/jfan.1997.3108. Google Scholar [3] K. T. Chen, Iterated path integrals,, Bull. Amer. Math. Soc., 83 (1977), 831. doi: 10.1090/S0002-9904-1977-14320-6. Google Scholar [4] W. Chen and M. Y. Chi, Hypoelliptic vector fields and almost periodic motions on the torus $T^n$,, Comm. Partial Differential Equations, 25 (2000), 337. doi: 10.1080/03605300008821516. Google Scholar [5] D. Damjanović, Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions,, J. Mod. Dyn., 1 (2007), 665. Google Scholar [6] D. Damjanović and A. Katok, Local rigidity of actions of higher rank abelian groups and KAM method,, Electron. Res. Announc. Amer. Math. Soc., 10 (2004), 142. doi: 10.1090/S1079-6762-04-00139-8. Google Scholar [7] D. Damjanović and A. Katok, Periodic cycle functionals and cocycle rigidity for certain partially hyperbolic $\bb R^k$ actions,, Discrete Contin. Dyn. Syst., 13 (2005), 985. doi: 10.3934/dcds.2005.13.985. Google Scholar [8] D. Damjanović and A. Katok, Local rigidity of restrictions of Weyl chamber flows,, C. R. Math. Acad. Sci. Paris, 344 (2007), 503. Google Scholar [9] R. de la Llave, J. M. Marco and R. Moriyón, Canonical perturbation theory of Anosov systems, and regularity results for the Livsic cohomology equation,, Bull. Amer. Math. Soc. (N.S.), 12 (1985), 91. Google Scholar [10] L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows,, Duke Math. J., 119 (2003), 465. doi: 10.1215/S0012-7094-03-11932-8. Google Scholar [11] L. Flaminio and G. Forni, On the cohomological equation for nilflows,, J. Mod. Dyn., 1 (2007), 37. Google Scholar [12] G. Forni, On the Greenfield-Wallach and Katok conjectures in dimension three,, in, 469 (2008), 197. Google Scholar [13] S. J. Greenfield and N. R. Wallach, Globally hypoelliptic vector fields,, Topology, 12 (1973), 247. doi: 10.1016/0040-9383(73)90011-6. Google Scholar [14] V. Guillemin and D. Kazhdan, On the cohomology of certain dynamical systems,, Topology, 19 (1980), 291. doi: 10.1016/0040-9383(80)90014-2. Google Scholar [15] V. Guillemin and D. Kazhdan, Some inverse spectral results for negatively curved $n$-manifolds,, in, XXXVI (1980), 153. Google Scholar [16] M. R. Herman, $L^2$ regularity of measurable solutions of a finite-difference equation of the circle,, Ergodic Theory Dynam. Systems, 24 (2004), 1277. doi: 10.1017/S0143385704000409. Google Scholar [17] S. Hurder, Problems on rigidity of group actions and cocycles,, Ergodic Theory Dynam. Systems, 5 (1985), 473. doi: 10.1017/S0143385700003084. Google Scholar [18] P. Iglesias, "Diffeology,", manuscript., (). Google Scholar [19] A. Katok, Cocycles, cohomology and combinatorial constructions in ergodic theory,, in, 69 (2001), 107. Google Scholar [20] A. Katok, Combinatorial constructions in ergodic theory and dynamics,, University Lecture Series, 30 (2003). Google Scholar [21] A. Kocsard, Cohomologically rigid vector fields: The Katok conjecture in dimension 3,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1165. doi: 10.1016/j.anihpc.2008.07.005. Google Scholar [22] R. U. Luz and N. M. dos Santos, Cohomology-free diffeomorphisms of low-dimension tori,, Ergodic Theory Dynam. Systems, 18 (1998), 985. doi: 10.1017/S0143385798108222. Google Scholar [23] S. Matsumoto, The parameter rigid flows on 3-manifolds,, in, 498 (2009), 135. Google Scholar [24] D. Mieczkowski, The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory,, J. Mod. Dyn., 1 (2007), 61. Google Scholar [25] F. R. Hertz and J. R. Hertz, Cohomology free systems and the first Betti number,, Discrete Contin. Dyn. Syst., 15 (2006), 193. doi: 10.3934/dcds.2006.15.193. Google Scholar [26] C. H. Taubes, The Seiberg-Witten equations and the Weinstein conjecture,, Geom. Topol., 11 (2007), 2117. doi: 10.2140/gt.2007.11.2117. Google Scholar
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