On smooth conjugacy of hyperbolic diffeomorphism germs (Q385937)

In this short paper, the authors deal with the characterization of the \(C^1\)~conjugacy classes of local diffeomorphisms with a hyperbolic fixed point.NEWLINENEWLINEMore concretely, the authors prove that, given \(F_A\) the germ of a diffeomorphism of the form \(F_A(x) = Ax + \tilde F_A(x)\), where \(x \in \mathbb{R}^n\), \(D\tilde F_A(0) = 0\), \(A\) is a hyperbolic linear map with eigenvalues \(\mu_1, \dots, \mu_s, \nu_1, \dots, \nu_u\), \(s+u= n\), satisfying \(|\mu_s| \leq|\mu_{s-1}| \leq \cdots |\mu_1| < 1 < |\nu_1| \leq \cdots \leq |\nu_u|\) NEWLINE\[NEWLINE \text{ and }\mu_{i_1}\nu_{j_1} \neq \mu_{i_2} \nu_{j_2}, \text{ if }(i_1,j_1) \neq (i_2,j_2),NEWLINE\]NEWLINE and \(\tilde F_A\) is generic, then \(F_A\) is \(C^1\)~conjugated to a hyperbolic germ \(F_B\) if and only if \(A\) and \(B\) are similar.NEWLINENEWLINEThe proof of the result is only sketched. It uses the Poincaré-Dulac normal form, the Samovol Theorem [\textit{G. Belitskii}, Russian Math. Surveys 33, No 1, 107--177 (1978; Zbl 0398.58009)] and transformations of Logarithmic Mourtada type [\textit{P. Bonckaert} et al., C. R., Math., Acad. Sci. Paris 336, No. 1, 19--22 (2003; Zbl 1036.34044)].