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

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 R^{n}, which vanish at this point. If S is a subset of R^{n}, containing the origin, we first define that: f, g ∈ m(n), satisfying f|_{S} = g|_{S}s are equivalent relative to R_{S} if there exists a germ of local diffeomorphism of R^{n}h, 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 = R^{n}_{_} = {(x_{1}, x_{2},...,x_{n}) ∈ R^{n} | x_{1} ≤ 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 R^{n}, 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) ∈ R^{2} | 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.

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PauloFerreiradaSilvaPortoJr_DO_1977.pdf (2.30 Mbytes)

Publishing Date

2022-07-05

CeTI-SC/STI

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