Le grand théorème de Picard pour les multifonctions analytiques finies
Volume 77 / 2001
Abstract
Let $D$ be a domain of the complex plane containing the origin. The famous great theorem of Émile Picard asserts that if $h$ is holomorphic on $D\setminus\{ 0\} $, with an essential singularity at 0, then the image under $h$ of any pointed neighbourhood of 0 covers all the complex plane, with at most one exception. Introducing the concept of essential singularity for analytic multifunctions, we extend this theorem to a finite analytic multifunction $K$, of degree $N$, defined on $D\setminus\{ 0\}$. In this case $\bigcup _{0 < |\lambda |< r}K (\lambda )$ covers all the complex plane, with at most $2N-1$ exceptions. In particular, this theorem can be used in the case of $N\times N$ matrices whose entries are holomorphic on $D\setminus \{ 0\} $ with essential singularities at 0. In this case, if their spectra avoid $2N$ points on a pointed neighbourhood of 0, these spectra must be constant.
