Stabilizers and orbits of smooth functions

Original · EN

Let f:Rᵐ → R be a smooth function such that f(0)=0. We give a condition on f when for arbitrary preserving orientation diffeomorphism ϕ:R → R such that ϕ(0)=0 the function ϕ∘ f is right equivalent to f, i.e. there exists a diffeomorphism h:Rᵐ → Rᵐ such that ϕ∘ f = f ∘ h at 0∈ Rᵐ. 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 → [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 that have compact support and leave [0,1] invariant. There are two natural right and left-right actions of Diff(M) and Diff(M) × Diff[⁰,¹](R) on C∞(M,R). Let Sₘ(f), SMR(f), Oₘ(f), and OMR(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 Sₘ(f) ≈ SMR(f) and Oₘ ≈ OMR. Similar results are obtained for smooth mappings M → S¹.