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 on a
Riemann surface we consider the
dynamical system generated by the
iterates of
denoted by
. We then call the
orbit of
as the set of forward iterates of
. We are interested in the asymptotic behavior of the orbits in
(which will usually be
, the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
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''. In this last case
where
is the
period and
means
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
''superattracting'' if
), ''repelling'' if
and indifferent if
. Indifferent periodic orbits can be either ''rationally indifferent'' or ''irrationally indifferent'', depending on whether
for some
or
for all
, 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
are analytically
conjugate to an irrational rotation of the complex unit disc.
Name
The disk is named in honor of
Carl Ludwig Siegel.
Gallery
SiegelDisk.jpg , Siegel disc for a polynomial-like mapping
FigureJuliaSetForPolynomialLike.jpg, Julia set for , where and is the golden ratio. Orbits of some points inside the Siegel disc emphasized
UnboundedSiegeldisk.jpg, Julia set for , where and is the golden ratio. Orbits of some points inside the Siegel disc emphasized. The Siegel disc is either unbounded or its boundary is an indecomposable continuum.
Golden Mean Quadratic Siegel Disc Speed.png , Filled Julia set for for Golden Mean rotation number with interior colored proportional to the average discrete velocity on the orbit = abs( z_(n+1) - z_n ). Note that there is only one Siegel disc and many preimages of the orbits within the Siegel disk
Quadratic Golden Mean Siegel Disc IIM.png
Quadratic Golden Mean Siegel Disc IIM Animated.gif
InfoldingSiegelDisk1over2.gif , Infolding Siegel disc near 1/2
InfoldingSiegelDisk1over3.gif, Infolding Siegel disc near 1/3. One can see virtual Siegel disc
InfoldingSiegelDisk2over7.gif, Infolding Siegel disc near 2/7
InfoldingSiegelDisk1over2animation.gif
Siegel disk for c = -0.749998153581339 +0.001569040474910 i.png
Julia set for fc(z) = z*z+c where c = -0.749998153581339 +0.001569040474910*I; t = 0.49975027919634618290 with orbits.png, Julia set for fc(z) = z*z+c where c = -0.749998153581339 +0.001569040474910*I. Internal angle in turns is t = 0.49975027919634618290
Siegel quadratic 3,2,1000,1... ,.png, Julia set of quadratic polynomial with Siegel disk for rotation number ,2,1000,1...
Siegel quadratic 3,2,1000,1... ,IIM.png
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
if there exists a biholomorphism
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.
[ Lennart Carleson and Theodore W. Gamelin, ''Complex Dynamics'', Springer 1993]
Later
Alexander D. Brjuno improved this condition on the irrationality, enlarging it to the
Brjuno number
In mathematics, a Brjuno number (sometimes spelled Bruno or Bryuno) is a special type of irrational number named for Russian mathematician Alexander Bruno, who introduced them in .
Formal definition
An irrational number \alpha is called a Brju ...
s.
[ (First appeared in 1990 as ]
Stony Brook IMS Preprint
, available a
arXiV:math.DS/9201272
)
This is part of the result from the
Classification of Fatou components.
See also
*
Douady rabbit
*
Herman ring
In the mathematical discipline known as complex dynamics, the Herman ring is a Fatou componentJohn Milnor''Dynamics in one complex variable'' Third Edition, Annals of Mathematics Studies, 160, Princeton Univ. Press, Princeton, NJ, 2006. where the ...
References
{{reflist
Siegel disks at Scholarpedia
Fractals
Limit sets
Complex dynamics