Stabilizers and orbits of smooth functions

Details Stabilizers and orbits of smooth functions Stabilizers and orbits of smooth functions

2004-11-27

arxiv.org/abs/math/0411612v3

Bulletin des Sciences Mathematiques, 130 (2006) 279-311

Mathematics

Functional Analysis

This is the second version of the paper. Now the cases P=R and P=S^1 considered from unique point of view. In particular, this

Scientific paper

10.1016/j.bulsci.2005.11.001

Let $f:R^m --> R$ be a smooth function such that $f(0)=0$. We give a condition on $f$ when for arbitrary preserving orientation diffeomorphism $\phi:R --> R$ such that $\phi(0)=0$ the function $\phi\circ f$ is right equivalent to $f$, i.e. there exists a diffeomorphism $h:R^m --> \R^m$ such that $\phi \circ f = f \circ h$ at $0\in R^m$. The requirement is that $\mrsfunc$ 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, $f:M --> [0,1]$ a surjective smooth function, $Diff(M)$ the group of diffeomorphisms of $M$, and $Diff^{[0,1]}(R)$ the group of diffeomorphisms of $R^1$ that have compact support and leave $[0,1]$ invariant. There are two natural right and left-right actions of $Diff(M)$ and $Diff(M) \times Diff^{[0,1]}(R)$ 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 $M-->S^1$.

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