∞-jets of difeomorphisms preserving orbits of vector fields

Original · EN

Let F be a smooth vector field defined in a neighborhood of the origin in Rⁿ, F(O)=0, and let Fₜ be its local flow. Denote by E the set of germs of diffeomorphisms h:Rⁿ → Rⁿ preserving orbits of F and let Eidʳ be the identity component of E with respect to Cʳ-topology. Then every Eidʳ contains a subset Sh consisting of mappings of the form Ff₍ₓ₎(x), where f: Rⁿ → R is a smooth function. It was proved earlier by the author that if F is a linear vector field, then Sh=Eid⁰. In this paper we present a class of vector fields for which Sh and Eid¹ coincide on the level of ∞-jets. We also establish a parameter rigidity of linear vector fields and "reduced" Hamiltonian vector fields of real homogeneous polynomials in two variables.