Stabilizers and orbits of smooth functions

Abstract

Let f : R m → R be a smooth function such that f ( 0 ) = 0 . We give a condition J(id) 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 : R m → R m such that ϕ ○ f = f ○ h at 0 ∈ 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, simple, and many other singularities. We also globalize this result as follows. Let M be a smooth compact manifold, f : M → [ 0 , 1 ] a surjective smooth function, D M the group of diffeomorphisms of M, and D R [ 0 , 1 ] 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 D M and D M × D R [ 0 , 1 ] on C ∞ ( M , R ) . Let S M ( f ) , S M R ( f ) , O M ( f ) , and O M R ( f ) be the corresponding stabilizers and orbits of f with respect to these actions. We prove that if f satisfies J(id) at each critical point and has additional mild properties, then the following homotopy equivalences hold: S M ( f ) ≈ S M R ( f ) and O M ( f ) ≈ O M R ( f ) . Similar results are obtained for smooth mappings M → S 1 .