Germs of mappings $\omega$-determined with respect to a~given group

G. R. Belitskii

Sbornik. Mathematics, Tome 23 (1974) no. 3, pp. 425-440

Résumé

Let $ J(n,p)$ be the space of germs of $C^\infty$-mappings $F\colon(R^n,0)\to(R^p,0)$ and $\mathfrak G$ a group operating on $J(n,p)$. The germ $F\in J(n,p)$ is called finitely determined with respect to $\mathfrak G$ if there exists an integer $k$ such that the orbit of the germ $F$ under the action of $\mathfrak G$ is uniquely determined by the $k$-jet of the germ $F$. The germ $F$ is called $\omega$-determined with respect to the group $\mathfrak G$ if each germ $G\in J(n,p)$ that has the same formal series as $F$ at the origin lies in the orbit of $F$ under the action of $\mathfrak G$. In this work, sufficient conditions are stated for $\omega$-determinedness. Examples are given of $\omega$-determined germs which are not finitely determined. Bibliography: 5 titles.