Sufficient conditions for local equivalence of mappings

Abstract

Let f, g : (R n , 0) → (R m , 0) be C r+1 mappings and let Z = {x ∈ R n : ν(df (x)) = 0}, 0 ∈ Z, m ≤ n. We will show that if there exist a neighbourhood U of 0 ∈ R n and constants C, C > 0 and k > 1 such that for x ∈ U ν(df (x)) ≥ C dist(x, Z) k−1 , |∂ s (fi − gi)(x)| ≤ C ν(df (x)) r+k−|s| , for any i ∈ {1,. .. , m} and for any s ∈ N n 0 such that |s| ≤ r, then there exists a C r diffeomorphism ϕ : (R n , 0) → (R n , 0) such that f = g • ϕ in a neighbourhood of 0 ∈ R n. By ν(df) we denote the Rabier function.