Connected components of partition preserving diffeomorphisms

TL;DR

This paper investigates the structure of the group of diffeomorphisms preserving a homogeneous polynomial in two variables, revealing conditions under which certain connected components coincide or differ.

Contribution

It characterizes the connected components of the group of partition-preserving diffeomorphisms for homogeneous polynomials in two variables.

Findings

$S(f, ext{infinity}) = \

$S(f,1) eq S(f,0)$ iff $f$ is a product of at least two distinct irreducible quadratic forms.

The equality of connected components $S(f, ext{infinity})$ through $S(f,1)$ for all such $f$.

Abstract

Let $f:\mathbb{R}^2 \to \mathbb{R}$ be a real homogeneous polynomial and $S(f)$ be the group of diffeomorphisms $h:\mathbb{R}^2 \to \mathbb{R}^2$ preserving $f$, i.e. $f \circ h = f$. Denote by $S(f,r)$, $(0\leq r \leq \infty)$, the identity path component of $S(f)$ with respect to the weak Whitney $C^{r}_{W}$-topology. We prove that $S(f,\infty) = \cdots = S(f,1)$ for all such $f$ and that $S(f,1) \not= S(f,0)$ if and only if $f$ is a product of at least two distinct irreducible over $\mathbb{R}$ quadratic forms.