Cohomologically Rigid Vector Fields : the Katok Conjecture in Dimension 3
Kocsard, Alejandro
Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009), p. 1165-1182 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU) 
@article{AIHPC_2009__26_4_1165_0,
     author = {Kocsard, Alejandro},
     title = {Cohomologically Rigid Vector Fields : the Katok Conjecture in Dimension 3},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {26},
     year = {2009},
     pages = {1165-1182},
     doi = {10.1016/j.anihpc.2008.07.005},
     mrnumber = {2542719},
     zbl = {1179.37039},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2009__26_4_1165_0}
}

Kocsard, Alejandro. Cohomologically Rigid Vector Fields : the Katok Conjecture in Dimension 3. Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) pp. 1165-1182. doi : 10.1016/j.anihpc.2008.07.005. http://gdmltest.u-ga.fr/item/AIHPC_2009__26_4_1165_0/

[1] Anosov D., Geodesic Flows on Closed Riemannian Manifolds of Negative Curvature, Proceedings of the Steklov Institute of Mathematics 90 (1967) 1-235. | MR 224110 | Zbl 0176.19101

[2] Bowen R., Walters P., Expansive One-Parameter Flows, Journal of Differential Equations 12 (1972) 180-193. | MR 341451 | Zbl 0242.54041

[3] Doering C., Persistently Transitive Vector Fields on Three-Dimensional Manifolds, in: Dynamical Systems and Bifurcation Theory, Pitman Res. Notes Math. Ser., vol. 160, Longman Sci. Tech., 1987. | MR 907891 | Zbl 0631.58016

[4] Flaminio L., Forni G., On the Cohomological Equation for Nilflows, Journal of Modern Dynamics 1 (1) (2007) 37-60. | MR 2261071 | Zbl 1114.37004

[5] Forni G., On the Greenfield-Wallach and Katok Conjectures, preprint, available at:, http://arxiv.org/abs/0706.3981.

[6] Fuller F., The Existence of Periodic Points, Annals of Mathematics 57 (2) (1953) 229-230. | MR 52764 | Zbl 0050.17203

[7] Greenfield S., Wallach N., Globally Hypoelliptic Vector Fields, Topology 12 (1973) 247-253. | MR 320502 | Zbl 0268.58007

[8] Herman M., Sur La Conjugaison Différentiable Des Difféomorphismes Du Cercle À Des Rotations, Publications Mathématiques de l'I.H.É.S. 49 (1979) 5-233. | Numdam | MR 538680 | Zbl 0448.58019

[9] Hurder S., Problems of Rigidity of Group Actions and Cocycles, Ergodic Theory & Dynamical Systems 5 (1985) 473-484. | MR 805843 | Zbl 0589.22006

[10] Katok A., Combinatorial Constructions in Ergodic Theory and Dynamical Systems, University Lecture Series, vol. 30, American Mathematical Society, 2003. | MR 2008435 | Zbl 1030.37001

[11] Katok A., Robinson E., Cocycles, Cohomology and Combinatorial Constructions in Ergodic Theory, in: Smooth Ergodic Theory and Its Applications, Proceedings of the Symposia in Pure Mathematics, vol. 69, 2001. | MR 1858535 | Zbl 0994.37003

[12] Katok A., Spatzier R., First Cohomology of Anosov Actions of Higher Rank Abelian Groups and Applications to Rigidity, Publications Mathématiques de l'I.H.É.S. 79 (1994) 131-156. | Numdam | MR 1307298 | Zbl 0819.58027

[13] A. Kocsard, Toward the classification of cohomology-free vector fields, Ph.D. thesis, IMPA, 2007.

[14] Mañé R., Quasi-Anosov Diffeomorphisms and Hyperbolic Manifolds, Transactions of the American Mathematical Society 229 (1977) 351-370. | MR 482849 | Zbl 0356.58009

[15] S. Matsumoto, The parameter rigid flows on 3-manifolds, preprint, 2007.

[16] Matsumoto S., Mitsumatsu Y., Leafwise Cohomology and Rigidity of Certain Lie Group Actions, Ergodic Theory & Dynamical Systems 23 (2003) 1839-1866. | MR 2032491 | Zbl 1052.22020

[17] Moser J., On the Volume Elements on a Manifold, Transactions of the American Mathematical Society 120 (1965) 186-294. | MR 182927 | Zbl 0141.19407

[18] Nakayama H., Noda T., Minimal Sets and Chain Recurrent Sets of Projective Flows Induced From Minimal Flows on 3-Manifolds, Discrete and Continuous Dynamical Systems 12 (4) (2005) 629-638. | MR 2129363 | Zbl 1075.37011

[19] Oseledets V., A Multiplicative Ergodic Theorem: Lyapunov Characteristic Numbers for Dynamical Systems, Transactions of the Moscow Mathematical Society 19 (1968) 197-231. | Zbl 0236.93034

[20] Paternain M., Expansive Flows and the Fundamental Group, Boletim da Sociedade Brasileira de Matemática 24 (2) (1993) 179-199. | MR 1254982 | Zbl 0796.58040

[21] Rodríguez Hertz F., Rodríguez Hertz J., Cohomology Free Systems and the First Betti Number, Discrete and Continuous Dynamical Systems 15 (2006) 193-196. | MR 2191392 | Zbl 1105.37013

[22] Rosenberg H., Roussarie R., Weil D., A Classification of Closed Orientable 3-Manifolds of Rank 2, Annals of Mathematics 91 (3) (1970) 449-464. | MR 270391 | Zbl 0195.25404

[23] Sacker R., Sell G., Existence of Dichotomies and Invariant Splittings for Linear Differential Equations, Journal of Differential Equations 15 (1974) 429-458. | MR 341458 | Zbl 0294.58008

[24] Santos N. D., Parameter Rigid Actions of the Heisenberg Group, Ergodic Theory & Dynamical Systems 27 (2007) 1719-1735. | MR 2371593 | Zbl 1127.37026

[25] Selgrade J., Isolated Invariant Sets for Flows on Vector Bundles, Transactions of the American Mathematical Society 203 (1975) 359-390. | MR 368080 | Zbl 0265.58004

[26] Selgrade J., Erratum to “isolated Invariant Sets for Flows on Vector Bundles”, Transaction of the American Mathematical Society 221 (1976) 249. | MR 413190 | Zbl 0329.58016

[27] Taubes C., The Seiberg-Witten Equations and the Weinstein Conjecture, preprint, available at:, http://arxiv.org/abs/math.SG/0611007. | MR 2350473 | Zbl 1135.57015

[28] Urzúa Luz R., The First Cohomology of Affine Z p -Actions on Tori and Applications to Rigidity, Bulletin of the Brazilian Mathematical Society 34 (2) (2003) 287-302. | MR 1992643 | Zbl 1037.37011

[29] Urzúa Luz R., Santos N. D., Cohomology Free Diffeomorphisms of Low-Dimension Tori, Ergodic Theory & Dynamical Systems 18 (1998) 985-1006. | MR 1645342 | Zbl 0922.58038

[30] Weinstein A., On the Hypotheses of Rabinowitz' Periodic Orbit Theorems, Journal of Differential Equations 33 (1979) 353-358. | MR 543704 | Zbl 0388.58020

[31] Yoccoz J.-C., Conjugaison Différentiable Des Difféomorphismes Du Cercle Dont Le Nombre De Rotation Vérifie Une Condition Diophantienne, Annales Scientifiques de l'É.N.S. 17 (1984) 333-359. | Numdam | MR 777374 | Zbl 0595.57027