Advanced Search

In this paper, we consider an analytic family of holomorphic mappings $f:M\times X\to X$ 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 $\mathrm{\Lambda }\left(\rho \right(m,.\left)\right)$ of $\rho (m,.)$ is of dimension $\ge 1$, we prove that $\mathrm{\Lambda }\left(\rho \right(m,.\left)\right)$ 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.

An example of a finite dimensional analytic space is exhibited, for which the Carathéodory integrated distance and the Carathéodory distance, although defining the same topology, are respectively complete and incomplete.