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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).

[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.

[3]

K. T. Chen, Iterated path integrals,, Bull. Amer. Math. Soc., 83 (1977), 831. 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. 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.

[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.

[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.

[8]

D. Damjanović and A. Katok, Local rigidity of restrictions of Weyl chamber flows,, C. R. Math. Acad. Sci. Paris, 344 (2007), 503.

[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.

[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.

[11]

L. Flaminio and G. Forni, On the cohomological equation for nilflows,, J. Mod. Dyn., 1 (2007), 37.

[12]

G. Forni, On the Greenfield-Wallach and Katok conjectures in dimension three,, in, 469 (2008), 197.

[13]

S. J. Greenfield and N. R. Wallach, Globally hypoelliptic vector fields,, Topology, 12 (1973), 247. 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. doi: 10.1016/0040-9383(80)90014-2.

[15]

V. Guillemin and D. Kazhdan, Some inverse spectral results for negatively curved $n$-manifolds,, in, XXXVI (1980), 153.

[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.

[17]

S. Hurder, Problems on rigidity of group actions and cocycles,, Ergodic Theory Dynam. Systems, 5 (1985), 473. doi: 10.1017/S0143385700003084.

[18]

P. Iglesias, "Diffeology,", manuscript., ().

[19]

A. Katok, Cocycles, cohomology and combinatorial constructions in ergodic theory,, in, 69 (2001), 107.

[20]

A. Katok, Combinatorial constructions in ergodic theory and dynamics,, University Lecture Series, 30 (2003).

[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.

[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.

[23]

S. Matsumoto, The parameter rigid flows on 3-manifolds,, in, 498 (2009), 135.

[24]

D. Mieczkowski, The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory,, J. Mod. Dyn., 1 (2007), 61.

[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.

[26]

C. H. Taubes, The Seiberg-Witten equations and the Weinstein conjecture,, Geom. Topol., 11 (2007), 2117. 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. doi: 10.1006/jfan.1997.3108.

[3]

K. T. Chen, Iterated path integrals,, Bull. Amer. Math. Soc., 83 (1977), 831. 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. 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.

[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.

[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.

[8]

D. Damjanović and A. Katok, Local rigidity of restrictions of Weyl chamber flows,, C. R. Math. Acad. Sci. Paris, 344 (2007), 503.

[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.

[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.

[11]

L. Flaminio and G. Forni, On the cohomological equation for nilflows,, J. Mod. Dyn., 1 (2007), 37.

[12]

G. Forni, On the Greenfield-Wallach and Katok conjectures in dimension three,, in, 469 (2008), 197.

[13]

S. J. Greenfield and N. R. Wallach, Globally hypoelliptic vector fields,, Topology, 12 (1973), 247. 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. doi: 10.1016/0040-9383(80)90014-2.

[15]

V. Guillemin and D. Kazhdan, Some inverse spectral results for negatively curved $n$-manifolds,, in, XXXVI (1980), 153.

[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.

[17]

S. Hurder, Problems on rigidity of group actions and cocycles,, Ergodic Theory Dynam. Systems, 5 (1985), 473. doi: 10.1017/S0143385700003084.

[18]

P. Iglesias, "Diffeology,", manuscript., ().

[19]

A. Katok, Cocycles, cohomology and combinatorial constructions in ergodic theory,, in, 69 (2001), 107.

[20]

A. Katok, Combinatorial constructions in ergodic theory and dynamics,, University Lecture Series, 30 (2003).

[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.

[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.

[23]

S. Matsumoto, The parameter rigid flows on 3-manifolds,, in, 498 (2009), 135.

[24]

D. Mieczkowski, The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory,, J. Mod. Dyn., 1 (2007), 61.

[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.

[26]

C. H. Taubes, The Seiberg-Witten equations and the Weinstein conjecture,, Geom. Topol., 11 (2007), 2117. doi: 10.2140/gt.2007.11.2117.

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