Earle–Hamilton fixed-point theorem
The Earle–Hamilton fixed-point theorem explains when a holomorphic map from a domain to itself in a complex Banach space has a unique fixed point, and when repeated application of the map brings any starting point to that fixed point.
Setup. Let D be a connected open subset of a complex Banach space X, and let f: D → D be holomorphic. By shrinking D if needed, we may assume D is bounded. The key idea is to use the Carathéodory metric on D, a way of measuring distance that comes from bounded holomorphic functions on D.
Main idea. In this metric, the map f acts as a contraction: it brings points closer together. Because of this contraction property, one can apply the Banach fixed-point theorem to f (restricted to the closure of f(D) with this metric). As a result, f has a unique fixed point x in D, and for any starting point y in D, the iterates fⁿ(y) converge to x.
Why it’s useful. The argument relies on a Schwarz-Pick–type inequality that shows how distances shrink under f. The Banach theorem then guarantees existence and uniqueness of the fixed point and convergence of iterates.
Extensions and related cases. In finite dimensions, fixed points often follow from Brouwer’s theorem without holomorphicity. The same contraction idea works in more geometric settings, such as bounded symmetric domains with the Bergman metric: these domains are complete under the Bergman metric, and the action of the complexified symmetry group is by contractions, so a fixed point exists by Banach’s theorem. This approach extends, with continuity, to some infinite-dimensional bounded symmetric domains, including the Siegel disk of certain Hilbert-Schmidt operators, where the Earle–Hamilton idea still applies.
This page was last edited on 2 February 2026, at 10:55 (CET).