Itérées d’une famille analytique d’applications holomorphes et points fixes sur un produit

Belkhchicha, Larbi ; Vigué, Jean-Pierre

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 14 (2003) no. 1, pp. 23-31

Résumé

In this paper, we consider an analytic family of holomorphic mappings $f : M \times X \rightarrow X$ and the sequence $f^{n}$ of iterates of $f$. If the sequence is not compactly divergent, there exists an unique retraction $\rho(m,.)$ adherent to the sequence $f^{n}(m,.)$. If $X$ is a strictly convex taut domain in $\mathbb{C}^{n}$ and if the image $\Lambda(\rho(m,.))$ of $\rho(m,.)$ is of dimension $\ge 1$, we prove that $\Lambda(\rho(m,.))$ does not depend from $m\in M$. We apply this result to the existence of fixed points of holomorphic mappings on the product of two bounded strictly convex domains.