Revistes Catalanes amb Accés Obert (RACO)

Attracting domains for semi-attractive transformations of $\mathbf{C}^p$

M. Hakim


Let $F$ be a germ of analytic transformation of $(\bold C^p,0)$. We say that $F$ is semi-attractive at the origin, if $F'_{(0)}$ has one eigenvalue equal to 1 and if the other ones are of modulus strictly less than 1. The main result is: either there exists a curve of fixed points, or $F-\operatorname{Id}$ has multiplicity $k$ and there exists a domain of attraction with $k-1$ petals. We study also the case where $F$ is a global isomorphism of $\bold C^2$ and $F-\operatorname{Id}$ has multiplicity $k$ at the origin. This work has been inspired by two papers: one of P. Fatou (1924) and the other one of T. Ueda (1986).

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