Seal disc

Seal disks of the illustration ${\displaystyle f(z)=z^{2}+e^{2\pi i{\frac {{\sqrt {5}}-1}{2}}}z}$

In mathematics , seal discs are a term from the theory of complex dynamic systems . They are components of the Fatou set on which the dynamics are conjugated to an irrational rotation.

definition

Let be a holomorphic map between Riemann surfaces. A connected component of the Fatou amount is sealing disk to when there is a biholomorphic map to the unit disc with there, so that an irrational rotation, so a is. ${\displaystyle f\colon S\to S}$ ${\displaystyle U}$ ${\displaystyle {\mathcal {F}}(f)}$ ${\displaystyle z_{0}\in U}$ ${\displaystyle \phi \colon U\to D}$ ${\displaystyle D}$ ${\displaystyle \phi (z_{0})=0}$ ${\displaystyle \phi f\phi ^{-1}}$ ${\displaystyle \phi f\phi ^{-1}(z)=e^{2\pi i\alpha }z}$ ${\displaystyle \alpha \in \mathbb {R} \setminus \mathbb {Q} }$

The question of whether there is a given and a seal disk is referred to in older literature as a center problem. ${\displaystyle f}$ ${\displaystyle z_{0}}$

Sentences by Siegel and Brjuno

In order for there to be a seal disk, the rotation number must be of the form with an irrational number . ${\displaystyle f^{\prime }(z_{0})}$ ${\displaystyle e^{2\pi i\alpha }}$ ${\displaystyle \alpha }$

Siegel proved in 1942 that there is a Siegel disc if one finds constants and , so that applies to all rational numbers . ${\displaystyle C>0}$ ${\displaystyle \nu \geq 2}$ ${\displaystyle \vert \alpha -{\frac {p}{q}}\vert \geq {\frac {C}{q^{\nu }}}}$ ${\displaystyle {\frac {p}{q}}}$

Rüßmann and Brjuno improved this arithmetic condition in the late 1960s.

Set of Brjuno : There is a seal disk when the result in the continued fraction expansion of the following applies: . (Such numbers are called Brjuno numbers .) ${\displaystyle {\frac {p_{n}}{q_{n}}}}$ ${\displaystyle \alpha }$ ${\displaystyle \sum _{n}{\frac {\log q_{n+1}}{q_{n}}}<\infty }$

Yoccoz proved in 1988 that Brjuno's condition is optimal. For every number that is not a Brjuno number there is a non-linearizable holomorphic function. ${\displaystyle \alpha }$ ${\displaystyle f(z)=z^{2}+e^{2\pi i\alpha }z}$

literature

• Carl Ludwig Siegel: Iteration of analytic functions , Ann. Math. 43, 607-612 (1942)
• Alexander D. Brjuno: Analytic form of differential equations. I, II , Trudy Moskovskogo Matematičeskogo Obščestva 25, 119–262 (1971)
• Jean-Christophe Yoccoz: Théorème de Siegel, nombres de Bruno et polynômes quadratiques , Petits diviseurs en dimension 1, Astérisque 231, 3-88 (1995)

Web links

• Scholarpedia: Siegel disks / Linearization