Siegel disc

Siegel disc is a connected component in the Fatou set where the dynamics is analytically conjugate to an irrational rotation.
Description.
Given a holomorphic endomorphism formula_1 on a Riemann surface formula_2 we consider the dynamical system generated by the iterates of formula_3 denoted by formula_4. We then call the orbit formula_5 of formula_6 as the set of forward iterates of formula_6. We are interested in the asymptotic behavior of the orbits in formula_2 (which will usually be formula_9, the complex plane or formula_10, the Riemann sphere), and we call formula_2 the phase plane or "dynamical plane".
One possible asymptotic behavior for a point formula_6 is to be a fixed point, or in general a "periodic point". In this last case formula_13 where formula_14 is the period and formula_15 means formula_6 is a fixed point. We can then define the "multiplier" of the orbit as formula_17 and this enables us to classify periodic orbits as "attracting" if formula_18 "superattracting" if formula_19), "repelling" if formula_20 and indifferent if formula_21. Indifferent periodic orbits can be either "rationally indifferent" or "irrationally indifferent", depending on whether formula_22 for some formula_23 or formula_24 for all formula_23, respectively.
Siegel discs are one of the possible cases of connected components in the Fatou set (the complementary set of the Julia set), according to Classification of Fatou components, and can occur around irrationally indifferent periodic points. The Fatou set is, roughly, the set of points where the iterates behave similarly to their neighbours (they form a normal family). Siegel discs correspond to points where the dynamics of formula_3 are analytically 
conjugate to an irrational rotation of the complex unit disc.
Name.
The disk is named in honor of Carl Ludwig Siegel.
Formal definition.
Let formula_27 be a holomorphic endomorphism where formula_2 is a Riemann surface, and let U be a connected component of the Fatou set formula_29. We say U is a Siegel disc of f around the point formula_6 if there exists a biholomorphism formula_31 where formula_32 is the unit disc and such that formula_33 for some formula_34 and formula_35.
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 Alexander 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.