Stabilizers and orbits of smooth functions

Abstract

Let $f:R^m \to R$ be a smooth function such that $f(0)=0$. We give a condition on $f$ when for arbitrary preserving orientation diffeomorphism $\phi:\mathbb{R} \to \mathbb{R}$ such that $\phi(0)=0$ the function $\phi\circ f$ is right equivalent to $f$, i.e. there exists a diffeomorphism $h:\mathbb{R}^m \to \mathbb{R}^m$ such that $\phi \circ f = f \circ h$ at $0\in \mathbb{R}^m$. The requirement is that $f$ belongs to its Jacobi ideal. This property is rather general: it is invariant with respect to the stable equivalence of singularities, and holds for non-degenerated critical points, simple singularities and many others. We also globalize this result as follows. Let $M$ be a smooth compact manifold, $f:M \to [0,1]$ a surjective smooth function, $\mathrm{Diff}(M)$ the group of diffeomorphisms of $M$, and $\mathrm{Diff}^{[0,1]}(\mathbb{R})$ the group of diffeomorphisms of $\mathbb{R}$ that have compact support and leave $[0,1]$ invariant. There are two natural right and left-right actions of $\mathrm{Diff}(M)$ and $\mathrm{Diff}(M) \times \mathrm{Diff}^{[0,1]}(\mathbb{R})$ on $C^{\infty}(M,R)$. Let $S_M(f)$, $S_{MR}(f)$, $O_{M}(f)$, and $O_{MR}(f)$ be the corresponding stabilizers and orbits of $f$ with respect to these actions. Under mild assumptions on $f$ we get the following homotopy equivalences $S_M(f) \approx S_{MR}(f)$ and $O_M \approx O_{MR}$. Similar results are obtained for smooth mappings $M \to S^1$.