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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 and Continuous Dynamical Systems, 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). [2] J. C. Baez and S. Sawin, Functional integration on spaces of connections, J. Funct. Anal., 150 (1997), 1-26. doi: 10.1006/jfan.1997.3108. [3] K. T. Chen, Iterated path integrals, Bull. Amer. Math. Soc., 83 (1977), 831-879. doi: 10.1090/S0002-9904-1977-14320-6. [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), 337-354. doi: 10.1080/03605300008821516. [5] D. Damjanović, Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions, J. Mod. Dyn., 1 (2007), 665-688. [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), 142-154 (electronic). doi: 10.1090/S1079-6762-04-00139-8. [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), 985-1005. doi: 10.3934/dcds.2005.13.985. [8] D. Damjanović and A. Katok, Local rigidity of restrictions of Weyl chamber flows, C. R. Math. Acad. Sci. Paris, 344 (2007), 503-508. [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), 91-94. [10] L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J., 119 (2003), 465-526. doi: 10.1215/S0012-7094-03-11932-8. [11] L. Flaminio and G. Forni, On the cohomological equation for nilflows, J. Mod. Dyn., 1 (2007), 37-60. [12] G. Forni, On the Greenfield-Wallach and Katok conjectures in dimension three, in "Geometric and Probabilistic Structures in Dynamics," Contemp. Math., vol. 469, Amer. Math. Soc., Providence, RI, (2008), 197-213. [13] S. J. Greenfield and N. R. Wallach, Globally hypoelliptic vector fields, Topology, 12 (1973), 247-254. doi: 10.1016/0040-9383(73)90011-6. [14] V. Guillemin and D. Kazhdan, On the cohomology of certain dynamical systems, Topology, 19 (1980), 291-299. doi: 10.1016/0040-9383(80)90014-2. [15] V. Guillemin and D. Kazhdan, Some inverse spectral results for negatively curved $n$-manifolds, in "Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979)," Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., (1980), 153-180. [16] M. R. Herman, $L^2$ regularity of measurable solutions of a finite-difference equation of the circle, Ergodic Theory Dynam. Systems, 24 (2004), 1277-1281. doi: 10.1017/S0143385704000409. [17] S. Hurder, Problems on rigidity of group actions and cocycles, Ergodic Theory Dynam. Systems, 5 (1985), 473-484. doi: 10.1017/S0143385700003084. [18] P. Iglesias, "Diffeology," manuscript. [19] A. Katok, Cocycles, cohomology and combinatorial constructions in ergodic theory, in "Smooth Ergodic Theory and its Applications (Seattle, WA, 1999)," Proc. Sympos. Pure Math., vol. 69, Amer. Math. Soc., Providence, RI, (2001), 107-173. (In collaboration with E. A. Robinson, Jr.). [20] A. Katok, Combinatorial constructions in ergodic theory and dynamics, University Lecture Series, vol. 30, American Mathematical Society, Providence, RI, 2003. [21] A. Kocsard, Cohomologically rigid vector fields: The Katok conjecture in dimension 3, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1165-1182. doi: 10.1016/j.anihpc.2008.07.005. [22] R. U. Luz and N. M. dos Santos, Cohomology-free diffeomorphisms of low-dimension tori, Ergodic Theory Dynam. Systems, 18 (1998), 985-1006. doi: 10.1017/S0143385798108222. [23] S. Matsumoto, The parameter rigid flows on 3-manifolds, in "Foliations, Geometry, and Topology: Paul Schweitzer Festschrift," Contemp. Math., vol. 498, Amer. Math. Soc., Providence, RI, (2009), 135-139. [24] D. Mieczkowski, The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory, J. Mod. Dyn., 1 (2007), 61-92. [25] F. R. Hertz and J. R. Hertz, Cohomology free systems and the first Betti number, Discrete Contin. Dyn. Syst., 15 (2006), 193-196. doi: 10.3934/dcds.2006.15.193. [26] C. H. Taubes, The Seiberg-Witten equations and the Weinstein conjecture, Geom. Topol., 11 (2007), 2117-2202. doi: 10.2140/gt.2007.11.2117.

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##### References:
 [1] A. Avila and A. Kocsard, Cohomological equations and invariant distributions for minimal circle diffeomorphisms, arXiv:1002.3392, (2010). [2] J. C. Baez and S. Sawin, Functional integration on spaces of connections, J. Funct. Anal., 150 (1997), 1-26. doi: 10.1006/jfan.1997.3108. [3] K. T. Chen, Iterated path integrals, Bull. Amer. Math. Soc., 83 (1977), 831-879. doi: 10.1090/S0002-9904-1977-14320-6. [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), 337-354. doi: 10.1080/03605300008821516. [5] D. Damjanović, Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions, J. Mod. Dyn., 1 (2007), 665-688. [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), 142-154 (electronic). doi: 10.1090/S1079-6762-04-00139-8. [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), 985-1005. doi: 10.3934/dcds.2005.13.985. [8] D. Damjanović and A. Katok, Local rigidity of restrictions of Weyl chamber flows, C. R. Math. Acad. Sci. Paris, 344 (2007), 503-508. [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), 91-94. [10] L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J., 119 (2003), 465-526. doi: 10.1215/S0012-7094-03-11932-8. [11] L. Flaminio and G. Forni, On the cohomological equation for nilflows, J. Mod. Dyn., 1 (2007), 37-60. [12] G. Forni, On the Greenfield-Wallach and Katok conjectures in dimension three, in "Geometric and Probabilistic Structures in Dynamics," Contemp. Math., vol. 469, Amer. Math. Soc., Providence, RI, (2008), 197-213. [13] S. J. Greenfield and N. R. Wallach, Globally hypoelliptic vector fields, Topology, 12 (1973), 247-254. doi: 10.1016/0040-9383(73)90011-6. [14] V. Guillemin and D. Kazhdan, On the cohomology of certain dynamical systems, Topology, 19 (1980), 291-299. doi: 10.1016/0040-9383(80)90014-2. [15] V. Guillemin and D. Kazhdan, Some inverse spectral results for negatively curved $n$-manifolds, in "Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979)," Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., (1980), 153-180. [16] M. R. Herman, $L^2$ regularity of measurable solutions of a finite-difference equation of the circle, Ergodic Theory Dynam. Systems, 24 (2004), 1277-1281. doi: 10.1017/S0143385704000409. [17] S. Hurder, Problems on rigidity of group actions and cocycles, Ergodic Theory Dynam. Systems, 5 (1985), 473-484. doi: 10.1017/S0143385700003084. [18] P. Iglesias, "Diffeology," manuscript. [19] A. Katok, Cocycles, cohomology and combinatorial constructions in ergodic theory, in "Smooth Ergodic Theory and its Applications (Seattle, WA, 1999)," Proc. Sympos. Pure Math., vol. 69, Amer. Math. Soc., Providence, RI, (2001), 107-173. (In collaboration with E. A. Robinson, Jr.). [20] A. Katok, Combinatorial constructions in ergodic theory and dynamics, University Lecture Series, vol. 30, American Mathematical Society, Providence, RI, 2003. [21] A. Kocsard, Cohomologically rigid vector fields: The Katok conjecture in dimension 3, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1165-1182. doi: 10.1016/j.anihpc.2008.07.005. [22] R. U. Luz and N. M. dos Santos, Cohomology-free diffeomorphisms of low-dimension tori, Ergodic Theory Dynam. Systems, 18 (1998), 985-1006. doi: 10.1017/S0143385798108222. [23] S. Matsumoto, The parameter rigid flows on 3-manifolds, in "Foliations, Geometry, and Topology: Paul Schweitzer Festschrift," Contemp. Math., vol. 498, Amer. Math. Soc., Providence, RI, (2009), 135-139. [24] D. Mieczkowski, The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory, J. Mod. Dyn., 1 (2007), 61-92. [25] F. R. Hertz and J. R. Hertz, Cohomology free systems and the first Betti number, Discrete Contin. Dyn. Syst., 15 (2006), 193-196. doi: 10.3934/dcds.2006.15.193. [26] C. H. Taubes, The Seiberg-Witten equations and the Weinstein conjecture, Geom. Topol., 11 (2007), 2117-2202. doi: 10.2140/gt.2007.11.2117.
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