• Previous Article
    Periodic solutions of parabolic problems with hysteresis on the boundary
  • DCDS Home
  • This Issue
  • Next Article
    Singularly perturbed degenerated parabolic equations and application to seabed morphodynamics in tided environment
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.

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.

[1]

Odo Diekmann, Francesca Scarabel, Rossana Vermiglio. Pseudospectral discretization of delay differential equations in sun-star formulation: Results and conjectures. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2575-2602. doi: 10.3934/dcdss.2020196

[2]

Livio Flaminio, Giovanni Forni. On the cohomological equation for nilflows. Journal of Modern Dynamics, 2007, 1 (1) : 37-60. doi: 10.3934/jmd.2007.1.37

[3]

Zhenqi Jenny Wang. The twisted cohomological equation over the geodesic flow. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 3923-3940. doi: 10.3934/dcds.2019158

[4]

James Tanis, Zhenqi Jenny Wang. Cohomological equation and cocycle rigidity of discrete parabolic actions. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 3969-4000. doi: 10.3934/dcds.2019160

[5]

Fedor Petrov, Zhi-Wei Sun. Proof of some conjectures involving quadratic residues. Electronic Research Archive, 2020, 28 (2) : 589-597. doi: 10.3934/era.2020031

[6]

Yunping Jiang. On a question of Katok in one-dimensional case. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1209-1213. doi: 10.3934/dcds.2009.24.1209

[7]

Michelle Nourigat, Richard Varro. Conjectures for the existence of an idempotent in $\omega $-polynomial algebras. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1543-1551. doi: 10.3934/dcdss.2011.4.1543

[8]

Xiaojun Huang, Jinsong Liu, Changrong Zhu. The Katok's entropy formula for amenable group actions. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4467-4482. doi: 10.3934/dcds.2018195

[9]

El Houcein El Abdalaoui, Joanna Kułaga-Przymus, Mariusz Lemańczyk, Thierry de la Rue. The Chowla and the Sarnak conjectures from ergodic theory point of view. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 2899-2944. doi: 10.3934/dcds.2017125

[10]

L. Bakker. The Katok-Spatzier conjecture, generalized symmetries, and equilibrium-free flows. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1183-1200. doi: 10.3934/cpaa.2013.12.1183

[11]

Giovanni Forni. The cohomological equation for area-preserving flows on compact surfaces. Electronic Research Announcements, 1995, 1: 114-123.

[12]

Bassam Fayad, Zhiyuan Zhang. An effective version of Katok's horseshoe theorem for conservative C2 surface diffeomorphisms. Journal of Modern Dynamics, 2017, 11: 425-445. doi: 10.3934/jmd.2017017

[13]

Flank D. M. Bezerra, Jacson Simsen, Mariza Stefanello Simsen. Convergence of quasilinear parabolic equations to semilinear equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3823-3834. doi: 10.3934/dcdsb.2020258

[14]

Sergi Simon. Linearised higher variational equations. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827

[15]

Min He. A class of integrodifferential equations and applications. Conference Publications, 2005, 2005 (Special) : 386-396. doi: 10.3934/proc.2005.2005.386

[16]

Evelyn Sander, E. Barreto, S.J. Schiff, P. So. Dynamics of noninvertibility in delay equations. Conference Publications, 2005, 2005 (Special) : 768-777. doi: 10.3934/proc.2005.2005.768

[17]

Christian Pötzsche. Dichotomy spectra of triangular equations. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 423-450. doi: 10.3934/dcds.2016.36.423

[18]

Wenxiong Chen, Congming Li, Jiuyi Zhu. Fractional equations with indefinite nonlinearities. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1257-1268. doi: 10.3934/dcds.2019054

[19]

Grégoire Allaire, Carlos Conca, Luis Friz, Jaime H. Ortega. On Bloch waves for the Stokes equations. Discrete and Continuous Dynamical Systems - B, 2007, 7 (1) : 1-28. doi: 10.3934/dcdsb.2007.7.1

[20]

Alessandro Fonda, Rafael Ortega. Positively homogeneous equations in the plane. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 475-482. doi: 10.3934/dcds.2000.6.475

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (97)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]