Hamiltonian vector fields of homogeneous polynomials in two variables

Abstract

Let $g:{\mathbb{R}}^2\to \mathbb{R}$ be a homogeneous polynomial of degree $p>1$, $G=(-g_y^{\prime },g_x^{\prime })$ be its Hamiltonian vector field, and $G_t$ be the local flow generated by $G$. Denote by $E(G,O)$ the space of germs of $C^{\infty }$ diffeomorphisms $({\mathbb{R}}^2,O)\to ({\mathbb{R}}^2,O)$ that preserve orbits of $G$. Let also $E_{\mathrm{id}}(G,O)$ be the identity component of $E(G,O)$ with respect to $C^1$-topology. Suppose that $g$ has no multiple prime factors. Then we prove that for every $h\in E_{\mathrm{id}}(G,O)$ there exists a germ of a smooth function $\alpha :{\mathbb{R}}^2\to \mathbb{R}$ at $O$ such that $h(z)=G_{\alpha (z)}(z)$.

26 pages, 3 figures. In version 2 the latter section is removed, since it is not included into original article

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