  We consider classes of diffeomorphisms of Euclidean space with partial
asymptotic expansions at infinity; the remainder term lies in a weighted
Sobolev space whose properties at infinity fit with the desired application. We
show that two such classes of asymptotic diffeomorphisms form topological
groups under composition. As such, they can be used in the study of fluid
dynamics according to the method of V. Arnold. Specific applications have been
obtained for the Camassa-Holm equation and the Euler equations.
