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A fixed-point theorem for holomorphic maps. (English) Zbl 0813.47068
Summary: We consider the action on the maximal ideal space $$M$$ of the algebra $$H$$ of bounded analytic functions, induced by an analytic self-map of a complex manifold $$X$$. After some general preliminaries, we focus on the question of the existence of fixed points for this action, in the case when $$X$$ is the open unit disk $$D$$. We classify the fixed-point-free Möbius transformations, and we show that for an arbitrary analytic map from $$\mathbb{D}$$ into itself, the induced map has a fixed point, or it restricts to a fixed-point-free Möbius map on some analytic disk contained in $$M$$.
##### MSC:
 47H10 Fixed-point theorems 46J15 Banach algebras of differentiable or analytic functions, $$H^p$$-spaces