Siegel disc

Description

Given a holomorphic endomorphism f:S\to S on a Riemann surface S we consider the dynamical system generated by the iterates of f denoted by f^n=f\circ\stackrel{\left(n\right)}{\cdots}\circ f. We then call the orbit \mathcal{O}^+(z_0) of z_0 as the set of forward iterates of z_0. We are interested in the asymptotic behavior of the orbits in S (which will usually be \mathbb{C}, the complex plane or \mathbb{\hat C}=\mathbb{C}\cup{\infty}, the Riemann sphere), and we call S the phase plane or dynamical plane.

One possible asymptotic behavior for a point z_0 is to be a fixed point, or in general a periodic point. In this last case f^p(z_0)=z_0 where p is the period and p=1 means z_0 is a fixed point. We can then define the multiplier of the orbit as \rho=(f^p)'(z_0) and this enables us to classify periodic orbits as attracting if |\rho| superattracting if |\rho|=0), repelling if |\rho|1 and indifferent if |\rho|=1. Indifferent periodic orbits can be either rationally indifferent or irrationally indifferent, depending on whether \rho^n=1 for some n\in\mathbb{Z} or \rho^n\neq1 for all 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 are analytically conjugate to an irrational rotation of the complex unit disc.

Name

The Siegel disc is named in honor of Carl Ludwig Siegel.

Gallery

SiegelDisk.jpg |Siegel disc for a polynomial-like mapping FigureJuliaSetForPolynomialLike.jpg|Julia set for B(z)=\lambda a(e^{z/a}(z+1-a)+a-1), where a=15-15i and \lambda is the golden ratio. Orbits of some points inside the Siegel disc emphasized UnboundedSiegeldisk.jpg|Julia set for B(z)=\lambda a(e^{z/a}(z+1-a)+a-1), where a=-0.33258+0.10324i and \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.

Golden Mean Quadratic Siegel Disc Speed.png | Filled Julia set for 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 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) = zz+c where c = -0.749998153581339 +0.001569040474910I; t = 0.49975027919634618290 with orbits.png|Julia set for fc(z) = zz+c where c = -0.749998153581339 +0.001569040474910I. 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 [3,2,1000,1...]

Siegel quadratic 3,2,1000,1... ,IIM.png

Formal definition

Let f\colon S\to S be a holomorphic endomorphism where S is a Riemann surface, and let U be a connected component of the Fatou set \mathcal{F}(f). We say U is a Siegel disc of f around the point z_0 if there exists a biholomorphism \phi:U\to\mathbb{D} where \mathbb{D} is the unit disc and such that \phi(f^n(\phi^{-1}(z)))=e^{2\pi i\alpha n}z for some \alpha\in\mathbb{R}\backslash\mathbb{Q} and \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 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.

References

• Siegel disks at Scholarpedia

References

1. [https://www.ibiblio.org/e-notes/MSet/Fagella.htm Polynomial-like maps by Nuria Fagella in The Mandelbrot and Julia sets Anatomy]
2. Rubén Berenguel and Núria Fagella ''An entire transcendental family with a persistent Siegel disc'', 2009 preprint: [https://arxiv.org/abs/0907.0116 arXiV:0907.0116]
3. [[Lennart Carleson]] and Theodore W. Gamelin, ''Complex Dynamics'', Springer 1993
4. Milnor, John W.. (2006-04-24). "Dynamics in One Complex Variable". [[ Princeton University Press]].