A Siegel disc or Siegel disk is a connected component in the Fatou set where the dynamics is analytically conjugate to an irrational rotation.
Given a holomorphic endomorphism
on a Riemann surface
we consider the dynamical system generated by the iterates of
We then call the orbit
as the set of forward iterates of
We are interested in the asymptotic behavior of the orbits in
, the complex plane or
, the Riemann sphere), and we call
the phase plane or dynamical plane.
One possible asymptotic behavior for a point
is to be a fixed point, or in general a periodic point.
is a fixed point.
We can then define the multiplier of the orbit as
and this enables us to classify periodic orbits as attracting if
Indifferent periodic orbits can be either rationally indifferent or irrationally indifferent, depending on whether
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
are analytically conjugate to an irrational rotation of the complex unit disc.
The Siegel disc is named in honor of Carl Ludwig Siegel.
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
if there exists a biholomorphism
is the unit disc and such that
2 π i α n
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.
[3] Later Alexander D. Brjuno improved this condition on the irrationality, enlarging it to the Brjuno numbers.
[4] This is part of the result from the Classification of Fatou components.