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 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 split in rationally indifferent and 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 conjugated to an irrational rotation of the complex disc.
Read more about this topic: Siegel Disc
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