Não disponível

We introduce here two equivalence relations for germs in m(n), the ideal of real differentiable function-germs at the origin of Rⁿ, which vanish at this point. If S is a subset of Rⁿ, containing the origin, we first define that: f, g ∈ m(n), satisfying f|_S = g|_Ss are equivalent relative to R_S if there exists a germ of local diffeomorphism of Rⁿh, at 0, such that: g = foh and hl_S = Id_S. If S = x R^n-s s ≥ 1, we define the finite determinacy relative to R_S in the obvious way. Following Mather ideas, we obtain some very interesting results where the main one states that: f ∈ m(n) is finitely determined relative to R_S if and only if it is (right) finitely determined. Setting S = Rⁿ_{_} = {(x₁, x₂,...,x_n) ∈ Rⁿ | x₁ ≤ O}, we say that f ∈ m(n) is S-stable if for any g ∈ m(n) such that gl_s = fl_S, then f and g are equivalent relative to R_S. We also give a sufficient condition for S-stability and prove that a (right) finitely determined function-germ is S-stable. However, we observe that finite determinacy is not a necessary condition for S-stability. After we define that f and g in m(n) are equivalent relative to R^*_S, if there exists a germ of local diffeomorphism h, at 0 of Rⁿ, which conjugates them and preserves S. In other words, h conjugates the given germs and satisfies: h(S) ⊂ S. Considering S = (0) x R^n-s, we state the finite determinacy relative to R^*_S, in the natural way. Our main result in this direction is: f ∈ m(n) is finitely determined relative to R^*_S if and only if f and f|_S are (right) finitely determined. Finally, setting S = {(x,y) ∈ R² | x = O}, we prove that: If f ∈ m(2) is a (right) finitely determined function, then the necessary and sufficient condition that f be finitely determined relative to R^*_S is that f(x,y) = 0 has .contacts of finite order with y-axis. We finish giving reduced forms (normal forms) for a Morse germ f &isisn; m(2) through diffeomorphism germs which lets invariant the y-axis.