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:
“True variety is in that plenitude of real and unexpected elements, in the branch charged with blue flowers thrusting itself, against all expectations, from the springtime hedge which seems already too full, while the purely formal imitation of variety ... is but void and uniformity, that is, that which is most opposed to variety....”
—Marcel Proust (1871–1922)
“... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
—Adrienne Rich (b. 1929)