Sur les points fixes et les cycles r\'epulsifs au voisinage d'une singularit\'e essentielle isol\'ee \`a l'instar de la m\'ethode de Zalcman

Abstract

Let $g$ be a holomorphic function in the neighbourhoods of an isolated essential singularity $v$: if $g$ omits a complex value there, then $v$ may be approached by a sequence of repelling fixed points for $g$, whose multipliers diverge to $\infty$. This implies that an entire function omitting a value or a non-M\"obius self-map of the punctured plane admit infinite repelling fixed points, whose multipliers diverge to $\infty$. By another point of view, we show that, if $v$ is not Picard-exceptional for $g$, then $v$ can be approached by a sequence of 2-cycles of $g$: these cycles are repelling if $v$ is not a completely branched value.