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Cohomology equations and commutators of germs of contact diffeomorphisms

Authors: Augustin Banyaga, Rafael de la Llave and C. Eugene Wayne
Journal: Trans. Amer. Math. Soc. 312 (1989), 755-778
MSC: Primary 58F05
MathSciNet review: 935530
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Abstract: We study the group of germs of contact diffeomorphisms at a fixed point. We prove that the abelianization of this group is isomorphic to the multiplicative group of real positive numbers. The principal ingredient in this proof is a version of the Sternberg linearization theorem in which the conjugating diffeomorphism preserves the contact structure.

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  • [AR] R. Abraham and J. Robbin, Transversal mappings and flows, Benjamin, New York, 1967. MR 0240836 (39:2181)
  • [BP] A. Banyaga and J. Pulido, On the group of contact diffeomorphisms of $ {{\mathbf{R}}^{2n + 1}}$, Bol. Soc. Mat. Mexicana 23 (1978), 43-47. MR 579661 (81m:58024)
  • [BLW] A. Banyaga, R. de La Llave and E. Wayne, Cohomology equations near hyperbolic points, and applications to normal forms of symplectic systems, in preparation.
  • [C] M. Chaperon, Géométrie différentielle et singularités de systèmes dynamiques, Astérisque 138-139 (1986).
  • [D] L. E. Dickson, Theory of linear groups in arbitrary field, Trans. Amer. Math. Soc. 2 (1901), 363-394. MR 1500573
  • [G] J. Gray, Some global properties of contact structures, Ann. of Math. 69 (1959), 421-450. MR 0112161 (22:3016)
  • [HS] P. Hilton and U. Stammback, Homological algebra, Graduate Texts in Math., no. 4, Springer-Verlag, Berlin, 1970.
  • [I] M. C. Irwin, On the smoothness of the composition map, Quart. J. Math. Oxford 2 (1972), 113-133. MR 0305434 (46:4564)
  • [L] H. B. Lawson, The qualitative theory of foliations, CBMS Regional Conf. Ser. in Math., no. 27, Amer. Math. Soc., Providence, R.I.
  • [Li] P. Liberman, Sur les automorphismes infinitesimaux des structures symplectiques et des structures de Contact, Colloque de Géométrie Différentelle Globale Bruxelles (1958), Louvain, 1959. MR 0119153 (22:9919)
  • [LMM] R. de la Llave, J. M. Marco and R. Moriyón, Canonical perturbation theory of Anosov systems and regularity results for the Livsic cohomology equation, Ann. of Math. 123 (1986), 537-611. MR 840722 (88h:58091)
  • [Ly] V. V. Lychagin, On sufficient orbits of a group of contact diffeomorphisms, Math. USSR-Sb. 33 (1977), 223-242. MR 0494278 (58:13184)
  • [Ly2] -, Local classification of non linear first order partial differential equations, Russian Math. Surveys 30 (1975), 105-175.
  • [MA] J. N. Mather, On the homology of Haefliger's classifying space, Differential Topology (V. Villani, ed.), C.I.M.E., V.III, Ciclo 1976, Varena, Italy.
  • [MD] D. McDuff, On the group of volume preserving diffeomorphisms of $ {{\mathbf{R}}^n}$, Trans. Amer. Math. Soc. 261 (1980), 103-113. MR 576866 (81k:58019)
  • [Ne] E. Nelson, Topics in dynamics. I: Flows, Princeton Univ. Press, Princeton, N.J., 1969. MR 0282379 (43:8091)
  • [P] J. Pulido, An equation of contact vector fields and the group of contact diffeomorphisms, Thesis, Princeton Univ., 1981.
  • [R] J. Robbin, On the existence theorem for differential equations, Proc. Amer. Math. Soc. 19 (1968), 1005-1006. MR 0227583 (37:3167)
  • [S] F. Sergeraert, Feuilletages et difféomorphismes infiniment tangents a l'identité, Invent. Math. 39 (1977), 253-275. MR 0474327 (57:13973)
  • [Se] G. Sell, Smooth linearization near a fixed point, Amer. J. Math. 107 (1985), 1035-1091. MR 805804 (87c:58095)
  • [St] S. Sternberg, On the structure of local homeomorphisms of Euclidean $ n$-space. II, Amer. J. Math. 80 (1958), 623-631. MR 0096854 (20:3336)
  • [St2] -, The structure of local homeomorphisms. III, Amer. J. Math. 81 (1959), 578-604. MR 0109853 (22:738)
  • [T] F. Takens, Normal forms for certain singularities of vector fields, Ann. Inst. Fourier (Grenoble) 23 (1973), 163-195. MR 0365620 (51:1872)
  • [Z] E. Zehnder, Generalized implicit functions theorems with applications to some small divisor problems. I, Comm. Pure Appl. Math. 28 (1975), 91-140; II, Comm. Pure Appl. Math. 29 (1976), 41--111. MR 0380867 (52:1764)
  • [Z2] -, A simple proof of a generalization of a theorem by C. L. Siegel in geometry and topology (Palis et al., ed.), Lecture Notes in Math., vol. 597, Springer-Verlag, New York, 1977. MR 0461575 (57:1560)

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