Area functionals and Godbillon-Vey cocycles
Annales de l'Institut Fourier, Volume 42 (1992) no. 1-2, pp. 421-447.

We investigate the natural domain of definition of the Godbillon-Vey 2- dimensional cohomology class of the group of diffeomorphisms of the circle. We introduce the notion of area functionals on a space of functions on the circle, we give a sufficiently large space of functions with nontrivial area functional and we give a sufficiently large group of Lipschitz homeomorphisms of the circle where the Godbillon-Vey class is defined.

Nous étudions le domaine naturel de définition de la classe de cohomologie de dimension 2 de Godbillon-Vey du groupe des difféomorphismes du cercle. On introduit la notion de fonctionnelle d’aire sur un espace de fonctions sur le cercle, on définit un espace suffisamment grand de fonctions sur le cercle avec fonctionnelle d’aire non triviale et on définit un groupe suffisamment grand d’homéomorphismes lipschitziens du cercle où la classe de Godbillon-Vey est définie.

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     title = {Area functionals and {Godbillon-Vey} cocycles},
     journal = {Annales de l'Institut Fourier},
     pages = {421--447},
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     address = {Grenoble},
     volume = {42},
     number = {1-2},
     year = {1992},
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Tsuboi, Takashi. Area functionals and Godbillon-Vey cocycles. Annales de l'Institut Fourier, Volume 42 (1992) no. 1-2, pp. 421-447. doi : 10.5802/aif.1298. http://www.numdam.org/articles/10.5802/aif.1298/

[1] R. Bott, On some formulas for the characteristic classes of group actions, Foliations and Gelfand-Fuks Cohomology, Proc. Rio de Janeiro, 1976, Lecture Notes in Math., Springer-Verlag, vol. 652 (1978). | Zbl

[2] G. Duminy et V. Sergiescu, Sur la nullité de l'invariant de Godbillon-Vey, C. R. Acad. Sci. Paris, 292 (1981), 821-824. | MR | Zbl

[3] E. Ghys, Sur l'invariance topologique de la classe de Godbillon-Vey, Ann. Inst. Fourier, 37-4 (1987), 59-76. | EuDML | Numdam | MR | Zbl

[4] E. Ghys, L'invariant de Godbillon-Vey, Seminaire Bourbaki, exposé n°706, 1988/1989. | EuDML | Numdam | Zbl

[5] E. Ghys et V. Sergiescu, Sur un groupe remarquable de difféomorphismes du cercle, Comment. Math. Helv., 62 (1987), 185-239. | EuDML | MR | Zbl

[6] C. Godbillon et J. Vey, Un invariant des feuilletages de codimension 1, C. R. Acad. Sci. Paris, 273 (1971), 92-95. | MR | Zbl

[7] P. Greenberg, Classifying spaces for foliations with isolated singularities, Transactions Amer. Math. Soc., 304 (1987), 417-429. | MR | Zbl

[8] M. Herman, Sur la conjugaison differentiable des difféomorphismes du cercle à des rotations, Publ. Math. I. H. E. S., 49 (1979), 5-234. | EuDML | Numdam | MR | Zbl

[9] S. Hurder and A. Katok, Differentiability, rigidity and Godbillon-Vey classes for Anosov flows, Publ. Math. I. H. E. S., 72 (1990), 2-61. | EuDML | Numdam | Zbl

[10] J. Mather, The vanishing of the homology of certain groups of homeomorphisms, Topology, 10 (1971), 297-298. | MR | Zbl

[11] J. Mather, Integrability in codimension 1, Comm. Math. Helv., 48 (1973), 195-233. | MR | Zbl

[12] J. Mather, Commutators of diffeomorphisms I, II and III, Comm. Math. Helv., 49 (1974), 512-528; 50 (1975), 33-40; 60 (1985), 122-124. | MR | Zbl

[13] J. Mather, On the homology of Haefliger's classifying space, Differential Topology, (1976), 71-116. | Zbl

[14] Y. Mitsumatsu, A relation between the topological invariance of the Godbillon-Vey invariant and the differentiability of Anosov foliations, Advanced Studies in Pure Math., 5, Foliations, (1985), 159-167. | MR | Zbl

[15] W. Thurston, Noncobordant foliations of S3, Bull. Amer. Math. Soc., 78 (1972), 511-514. | MR | Zbl

[16] W. Thurston, Foliations and groups of diffeomorphism, Bull. Amer. Math. Soc., 80 (1974), 304-307. | MR | Zbl

[17] T. Tsuboi, On the homology of classifying spaces for foliated products, Advanced Studies in Pure Math., 5, Foliations, (1985), 37-120. | MR | Zbl

[18] T. Tsuboi, On the foliated products of class C1, Annals of Math., 130 (1989), 227-271. | MR | Zbl

[19] T. Tsuboi, On the Hurder-Katok extension of the Godbillon-Vey invariant, J. of Fac. Sci., Univ. of Tokyo, Sec. IA, 37 (1990), 255-262. | MR | Zbl

[20] T. Tsuboi, Homological and dynamical study on certain groups of Lipschitz homeomorphisms of the circle, preprint. | Zbl

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