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 $\varphi :\mathbb{R}\to \mathbb{R}$ such that $\varphi (0)=0$ the function $\varphi \circ f$ is right equivalent to $f$, i.e. there exists a diffeomorphism $h:{\mathbb{R}}^m\to {\mathbb{R}}^m$ such that $\varphi \circ f=f\circ h$ 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 $M\to S^1$.

This is the second version of the paper. Now the cases $P=\mathbb{R}$ and $P=S^1$ considered from unique point of view. In particular, this covers the results of the preprint http://xxx.lanl.gov/math.FA/0503734. Moreover, the role of the condition for a function to belong to its Jacobi ideal is explained

Cited by in corpus (10)

• Homotopy types of diffeomorphism groups of polar Morse-Bott foliations on lens spaces, 1
• Foliations with all non-closed leaves on non-compact surfaces
• Symmetries of degenerate center singularities of plane vector fields
• Image of a shift map along the orbits of a flow
• Connected components of partition preserving diffeomorphisms
• Homotopy types of diffeomorphisms groups of simplest Morse-Bott foliations on lens spaces, 2
• Functions with isolated singularities on surfaces, II
• One-dimensional foliations on topological manifolds
• Deformations of functions on surfaces
• Stabilizers and orbits of circle-valued smooth functions