Siegel disc

Siegel disc is a connected component in the Fatou set where the dynamics is analytically conjugated to an irrational rotation.

Given a holomorphic endomorphism



f

:

S

S

{\displaystyle f:S\to S}

on a Riemann surface



S

{\displaystyle S}

we consider the dynamical system generated by the iterates of



f

{\displaystyle f}

denoted by



f

n

=

f

(

n

)

f

{\displaystyle f^{n}=f\circ {\stackrel {\left(n\right)}{\cdots }}\circ f}

. We then call the orbit



O

+

(

z

0

)

{\displaystyle {\mathcal {O}}^{+}(z_{0})}

of



z

0

{\displaystyle z_{0}}

as the set of forward iterates of

z

0

{\displaystyle z_{0}}

. We are interested in the asymptotic behavior of the orbits in



S

{\displaystyle S}

(which will usually be



C

{\displaystyle \mathbb {C} }

, the complex plane or



C

^

=

C

{

}

{\displaystyle \mathbb {\hat {C}} =\mathbb {C} \cup \{\infty \}}

, the Riemann sphere), and we call



S

{\displaystyle S}

the phase plane or dynamical plane.

One possible asymptotic behavior for a point



z

0

{\displaystyle z_{0}}

is to be a fixed point, or in general a periodic point. In this last case



f

p

(

z

0

)

=

z

0

{\displaystyle f^{p}(z_{0})=z_{0}}

where



p

{\displaystyle p}

is the period and



p

=

1

{\displaystyle p=1}

means



z

0

{\displaystyle z_{0}}

is a fixed point. We can then define the multiplier of the orbit as



ρ

=

(

f

p

)

(

z

0

)

{\displaystyle \rho =(f^{p})'(z_{0})}



|

ρ

|

<

1

{\displaystyle |\rho |<1}

superattracting if



|

ρ

|

=

0

{\displaystyle |\rho |=0}

), repelling if



|

ρ

|

>

1

{\displaystyle |\rho |>1}



ρ

=

1

{\displaystyle \rho =1}

. Indifferent periodic orbits split in rationally indifferent and irrationally indifferent, depending on whether



ρ

n

=

1

{\displaystyle \rho ^{n}=1}

for some



n

Z

{\displaystyle n\in \mathbb {Z} }

or



ρ

n

1

{\displaystyle \rho ^{n}\neq 1}

for all



n

Z

{\displaystyle n\in \mathbb {Z} }

, 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



f

{\displaystyle f}



B

(

z

)

=

λ

a

(

e

z

/

a

(

z

+

1

a

)

+

a

1

)

{\displaystyle B(z)=\lambda a(e^{z/a}(z+1-a)+a-1)}

, where



a

=

15

15

i

{\displaystyle a=15-15i}

and



λ

{\displaystyle \lambda }

is the golden ratio. Orbits of some points inside the Siegel disc emphasized

Julia set for



B

(

z

)

=

λ

a

(

e

z

/

a

(

z

+

1

a

)

+

a

1

)

{\displaystyle B(z)=\lambda a(e^{z/a}(z+1-a)+a-1)}

, where



a

=

0.33258

+

0.10324

i

{\displaystyle a=-0.33258+0.10324i}

and



λ

{\displaystyle \lambda }

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.

Filled Julia set for



f

c

(

z

)

=

z

z

+

c

{\displaystyle f_{c}(z)=z*z+c}

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

Filled Julia set for



f

c

(

z

)

=

z

z

+

c

{\displaystyle f_{c}(z)=z*z+c}

for Golden Mean rotation number with Siegel disc and some orbits inside

Julia set of quadratic polynomial with Siegel disk for rotation number [3,2,1000,1. tenderize meat without a mallet..]

Infolding Siegel disc near 1/3. One can see virtual Siegel disc

Infolding Siegel disc near 2/7.

Let



f

:

S

S

{\displaystyle f:S\to S}

be a holomorphic endomorphism where



S

{\displaystyle S}

is a Riemann surface, and let U be a connected component of the Fatou set



F

(

f

)

{\displaystyle {\mathcal {F}}(f)}

. We say U is a Siegel disc of f around the point



z

0

{\displaystyle z_{0}}

if there exists a biholomorphism



ϕ

:

U

D

{\displaystyle \phi :U\to \mathbb {D} }

where



D

{\displaystyle \mathbb {D} }

is the unit disc and such that



ϕ

(

f

n

(

ϕ

1

(

z

)

)

)

=

e

2

π

i

α

n

z

{\displaystyle \phi (f^{n}(\phi ^{-1}(z)))=e^{2\pi i\alpha n}z}

for some



α

R

Q

{\displaystyle \alpha \in \mathbb {R} \backslash \mathbb {Q} }

and



ϕ

(

z

0

)

=

0

{\displaystyle \phi (z_{0})=0}

.

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.