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.

In this article we prove the existence of a new family of periodic solutions for discrete, nonlinear Schrödinger equations subject to spatially localized driving and damping and we show numerically that they provide a more accurate approximation to metastable states in these systems than previous proposals. We also study the stability properties of these solutions and show that they fit well with a previously proposed mechanism for the emergence and persistence of metastable behavior.

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