Cohomology equations and commutators of germs of contact diffeomorphisms

Banyaga, Augustin; de la Llave, Rafael; Wayne, C. Eugene

doi:10.1090/S0002-9947-1989-0935530-7

Cohomology equations and commutators of germs of contact diffeomorphisms
HTML articles powered by AMS MathViewer

by Augustin Banyaga, Rafael de la Llave and C. Eugene Wayne PDF

Trans. Amer. Math. Soc. 312 (1989), 755-778 Request permission

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.

References

Ralph Abraham and Joel Robbin, Transversal mappings and flows, W. A. Benjamin, Inc., New York-Amsterdam, 1967. An appendix by Al Kelley. MR 0240836
Augustin Banyaga and Javier Pulido, On the group of contact diffeomorphisms of $\textbf {R}^{2n+1}$, Bol. Soc. Mat. Mexicana (2) 23 (1978), no. 2, 43–47. MR 579661

Cohomology equations near hyperbolic points, and applications to normal forms of symplectic systems

Géométrie différentielle et singularités de systèmes dynamiques

138-139

Leonard Eugene Dickson, Theory of linear groups in an arbitrary field, Trans. Amer. Math. Soc. 2 (1901), no. 4, 363–394. MR 1500573, DOI 10.1090/S0002-9947-1901-1500573-3
John W. Gray, Some global properties of contact structures, Ann. of Math. (2) 69 (1959), 421–450. MR 112161, DOI 10.2307/1970192

Homological algebra

M. C. Irwin, On the smoothness of the composition map, Quart. J. Math. Oxford Ser. (2) 23 (1972), 113–133. MR 305434, DOI 10.1093/qmath/23.2.113

The qualitative theory of foliations

Paulette Libermann, Sur les automorphismes infinitésimaux des structures symplectiques et des structures de contact, Colloque Géom. Diff. Globale (Bruxelles, 1958) Centre Belge Rech. Math., Louvain, 1959, pp. 37–59 (French). MR 0119153
R. de la Llave, J. M. Marco, and R. Moriyón, Canonical perturbation theory of Anosov systems and regularity results for the Livšic cohomology equation, Ann. of Math. (2) 123 (1986), no. 3, 537–611. MR 840722, DOI 10.2307/1971334
V. V. Lyčagin, Sufficient orbits of a group of contact diffeomorphisms, Mat. Sb. (N.S.) 104(146) (1977), no. 2(10), 248–270, 335 (Russian). MR 0494278

Local classification of non linear first order partial differential equations

30

On the homology of Haefliger’s classifying space

Dusa McDuff, On the group of volume-preserving diffeomorphisms of $\textbf {R}^{n}$, Trans. Amer. Math. Soc. 261 (1980), no. 1, 103–113. MR 576866, DOI 10.1090/S0002-9947-1980-0576866-3
Edward Nelson, Topics in dynamics. I: Flows, Mathematical Notes, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1969. MR 0282379

An equation of contact vector fields and the group of contact diffeomorphisms

Joel W. Robbin, On the existence theorem for differential equations, Proc. Amer. Math. Soc. 19 (1968), 1005–1006. MR 227583, DOI 10.1090/S0002-9939-1968-0227583-0
F. Sergeraert, Feuilletages et difféomorphismes infiniment tangents à l’identité, Invent. Math. 39 (1977), no. 3, 253–275 (French). MR 474327, DOI 10.1007/BF01402976
George R. Sell, Smooth linearization near a fixed point, Amer. J. Math. 107 (1985), no. 5, 1035–1091. MR 805804, DOI 10.2307/2374346
Shlomo Sternberg, On the structure of local homeomorphisms of euclidean $n$-space. II, Amer. J. Math. 80 (1958), 623–631. MR 96854, DOI 10.2307/2372774
Shlomo Sternberg, The structure of local homeomorphisms. III, Amer. J. Math. 81 (1959), 578–604. MR 109853, DOI 10.2307/2372915
Floris Takens, Normal forms for certain singularities of vectorfields, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 2, 163–195 (English, with French summary). MR 365620
E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems. I, Comm. Pure Appl. Math. 28 (1975), 91–140. MR 380867, DOI 10.1002/cpa.3160280104
Eduard Zehnder, A simple proof of a generalization of a theorem by C. L. Siegel, Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976) Lecture Notes in Math., Vol. 597, Springer, Berlin, 1977, pp. 855–866. MR 0461575