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 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 is analytically conjugate to an irrational rotation of the complex disc.
The disk is named in honor of Carl Ludwig Siegel.
Julia set for , where and is the golden ratio. Orbits of some points inside the Siegel disc emphasized
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
Filled Julia set for for Golden Mean rotation number with Siegel disc and some orbits inside
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.
This is part of the result from the Classification of Fatou components.
|Wikibooks has a book on the topic of: Fractals/Iterations in the complex plane/siegel|
- Rubén Berenguel and Núria Fagella An entire transcendental family with a persistent Siegel disc, 2009 preprint: arXiV:0907.0116
- Lennart Carleson and Theodore W. Gamelin, Complex Dynamics, Springer 1993
- John W. Milnor, Dynamics in One Complex Variable (Third Edition), Annals of Mathematics Studies 160, Princeton University Press 2006 (First appeared in 1990 as a Stony Brook IMS Preprint Archived 2006-04-24 at the Wayback Machine., available as arXiV:math.DS/9201272.)