• • • • • • • • • • Doctoral Thesis
DOI
https://doi.org/10.11606/T.55.1977.tde-05072022-110523
Document
Author
Full name
Paulo Ferreira da Silva Porto Junior
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 1977
Supervisor
Committee
Loibel, Gilberto Francisco (President)
Barone Netto, Angelo
Favaro, Luiz Antonio
Ruzante, Auster
Teixeira, Marco Antonio
Title in Portuguese
DETERMINAÇÃO FINITA E ESTABILIDADE RELATIVAS DE GERMES DE FUNÇÕES
Keywords in Portuguese
Não disponível
Abstract in Portuguese
Não disponível
Title in English
Not available
Keywords in English
Not available
Abstract in English
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 RS if there exists a germ of local diffeomorphism of Rnh, at 0, such that: g = foh and hlS = IdS. If S = x Rn-s s ≥ 1, we define the finite determinacy relative to RS 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 RS if and only if it is (right) finitely determined. Setting S = Rn_ = {(x1, x2,...,xn) ∈ Rn | x1 ≤ O}, we say that f ∈ m(n) is S-stable if for any g ∈ m(n) such that gls = flS, then f and g are equivalent relative to RS. 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 Rn-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.