\infty-jets of difeomorphisms preserving orbits of vector fields

Details \infty-jets of difeomorphisms preserving orbits of vector fields \infty-jets of difeomorphisms preserving orbits of vector fields

2007-08-06

arxiv.org/abs/0708.0737v6

Mathematics

Dynamical Systems

34 pages. version 5. Many misprints are corrected and some minor changes are made

Scientific paper

Let F be a smooth vector field defined in a neighborhood of the origin in R^n, F(O)=0, and let F_t be its local flow. Denote by E the set of germs of diffeomorphisms h:R^n --> R^n preserving orbits of F and let E_{id}^r be the identity component of E with respect to C^r-topology. Then every E_{id}^{r} contains a subset Sh consisting of mappings of the form F_{f(x)}(x), where f: R^n --> R is a smooth function. It was proved earlier by the author that if F is a linear vector field, then Sh=E_{id}^0. In this paper we present a class of vector fields for which Sh and E_{id}^1 coincide on the level of \infty-jets. We also establish a parameter rigidity of linear vector fields and "reduced" Hamiltonian vector fields of real homogeneous polynomials in two variables.

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