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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

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 & 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). Google Scholar

[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.  Google Scholar

[3]

K. T. Chen, Iterated path integrals, Bull. Amer. Math. Soc., 83 (1977), 831-879. 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), 337-354. 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), 665-688.  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), 142-154 (electronic). 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), 985-1005. 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), 503-508.  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), 91-94.  Google Scholar

[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.  Google Scholar

[11]

L. Flaminio and G. Forni, On the cohomological equation for nilflows, J. Mod. Dyn., 1 (2007), 37-60.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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), 1277-1281. doi: 10.1017/S0143385704000409.  Google Scholar

[17]

S. Hurder, Problems on rigidity of group actions and cocycles, Ergodic Theory Dynam. Systems, 5 (1985), 473-484. 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 "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.).  Google Scholar

[20]

A. Katok, Combinatorial constructions in ergodic theory and dynamics, University Lecture Series, vol. 30, American Mathematical Society, Providence, RI, 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), 1165-1182. 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), 985-1006. doi: 10.1017/S0143385798108222.  Google Scholar

[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. 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), 61-92.  Google Scholar

[25]

F. R. Hertz and J. R. Hertz, Cohomology free systems and the first Betti number, Discrete Contin. Dyn. Syst., 15 (2006), 193-196. 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), 2117-2202. doi: 10.2140/gt.2007.11.2117.  Google Scholar

show all references

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), 1-26. doi: 10.1006/jfan.1997.3108.  Google Scholar

[3]

K. T. Chen, Iterated path integrals, Bull. Amer. Math. Soc., 83 (1977), 831-879. 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), 337-354. 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), 665-688.  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), 142-154 (electronic). 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), 985-1005. 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), 503-508.  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), 91-94.  Google Scholar

[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.  Google Scholar

[11]

L. Flaminio and G. Forni, On the cohomological equation for nilflows, J. Mod. Dyn., 1 (2007), 37-60.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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), 1277-1281. doi: 10.1017/S0143385704000409.  Google Scholar

[17]

S. Hurder, Problems on rigidity of group actions and cocycles, Ergodic Theory Dynam. Systems, 5 (1985), 473-484. 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 "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.).  Google Scholar

[20]

A. Katok, Combinatorial constructions in ergodic theory and dynamics, University Lecture Series, vol. 30, American Mathematical Society, Providence, RI, 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), 1165-1182. 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), 985-1006. doi: 10.1017/S0143385798108222.  Google Scholar

[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. 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), 61-92.  Google Scholar

[25]

F. R. Hertz and J. R. Hertz, Cohomology free systems and the first Betti number, Discrete Contin. Dyn. Syst., 15 (2006), 193-196. 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), 2117-2202. doi: 10.2140/gt.2007.11.2117.  Google Scholar

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