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.
Read more about this topic: Siegel Disc
Famous quotes containing the words formal and/or definition:
“Then the justice,
In fair round belly with good capon lined,
With eyes severe and beard of formal cut,
Full of wise saws and modern instances;
And so he plays his part.”
—William Shakespeare (1564–1616)
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)