Cohomology Equations and Commutators of Germs of Contact Diffeomorphisms

Abstract

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.

Homology of the Group of Leaf Preserving Homeomorphisms

“…The Poisson and Jacobi structures exhibit important examples of such structures but the problem of perfectness of their automorphism groups seems to be very difficult (see [1], [2] where only the transitive case of such a problem has been investigated).…”

The identity component of the leaf preserving diffeomorphism group is perfect

“…Also some related results concerning the groups of germs of diffeomorphisms are known, e.g. [3], [20].…”