Linearization of cohomology-free vector fields

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

Linearization of cohomology-free vector fields

Livio Flaminio ^1, and Miguel Paternain ^2,

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

Abstract
Related Papers

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.

Keywords: Cohomological Equations, Greenfield-Wallach and Katok conjectures..

Mathematics Subject Classification: Primary: 37-XX, 37C15, 37C4.

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.

show all references

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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