Siegel Disc - Formal Definition

Formal Definition

Let be a holomorphic endomorphism where is a Riemann surface, and let U be a connected component of the Fatou set . We say U is a Siegel disc of f around the point z_0 if there exists an analytic homeomorphism where is the unit disc and such that for some and .

Siegel's theorem proves the existence of Siegel discs for irrational numbers satisfying a strong irrationality condition (a Diophantine condition), thus solving an open problem since Fatou conjectured his theorem on the Classification of Fatou components.

Later A. D. Brjuno improved this condition on the irrationality, enlarging it to the Brjuno numbers.

This is part of the result from the Classification of Fatou components.