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

Abstract

Let $F$ be a smooth vector field defined in a neighborhood of the origin in ${\mathbb{R}}^n$, $F(O)=0$, and let $F_t$ be its local flow. Denote by $E$ the set of germs of diffeomorphisms $h:{\mathbb{R}}^n\to {\mathbb{R}}^n$ preserving orbits of $F$ and let $E_{\mathrm{id}}^r$ be the identity component of $E$ with respect to $C^r$-topology. Then every $E_{\mathrm{id}}^r$ contains a subset $Sh$ consisting of mappings of the form $F_{f(x)}(x)$, where $f:{\mathbb{R}}^n\to \mathbb{R}$ is a smooth function. It was proved earlier by the author that if $F$ is a linear vector field, then $Sh=E_{\mathrm{id}}^0$. In this paper we present a class of vector fields for which $Sh$ and $E_{\mathrm{id}}^1$ 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.

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

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