Itération des polynômes dans une algèbre de Banach

Abstract

Let $A$ be a complex Banach algebra, $f:A\rightarrow A$ be a polynomial function with coefficients in $A$. We define the Julia set of f, denoted by J(f). We give conditions which often determine in which set, $J(f)$ or $A\setminus J(f)$, the periodic point lies. We show that the closure of repelling periodic points is not always equals the Julia set. However, we use the theorem of Gelfand-Mazur to characterize the algebras where it's true.