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

TL;DR

This paper investigates the structure of diffeomorphisms preserving orbits of vector fields, showing that for certain classes, the set of orbit-preserving diffeomorphisms aligns with those generated by the flow, especially at the level of infinite jets.

Contribution

The paper identifies classes of vector fields where the set of orbit-preserving diffeomorphisms matches the flow-generated subset at the infinite jet level, extending previous results for linear fields.

Findings

For certain vector fields, $Sh$ and $E_{ ext{id}}^1$ coincide on $ ext{infty}$-jets.

Established parameter rigidity for linear and reduced Hamiltonian vector fields.

Extended the understanding of orbit-preserving diffeomorphisms beyond linear 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.