Herman ring

In mathematics , Herman rings (also Arnold-Herman rings ) are a term from the theory of complex dynamic systems . They are ring-shaped components of the Fatou set on which the dynamics are conjugated to an irrational rotation.

definition

Herman rings of the figure , where it was chosen so that the number of rotations is on the unit circle . ${\displaystyle f(z)={\frac {e^{2\pi i\tau }z^{2}(z-4)}{1-4z}}}$ ${\displaystyle \tau =0,6151732\ldots }$ ${\displaystyle f}$ ${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$

Let be a holomorphic map between Riemann surfaces. ${\displaystyle f\colon S\to S}$

A connected component of the Fatou set is called Herman’s ring if it is a biholomorphic map ${\displaystyle U}$ ${\displaystyle {\mathcal {F}}(f)}$

${\displaystyle \phi \colon U\to R}$

on a ring area

${\displaystyle R=\left\{z\in \mathbb {C} \colon r<\vert z\vert <1\right\}}$

with there, so an irrational twist, so ${\displaystyle r>0}$ ${\displaystyle \phi f\phi ^{-1}}$

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

for one is. ${\displaystyle \alpha \in \mathbb {R} \setminus \mathbb {Q} }$

history

After Fatous classification invariant components of the Fatou amount for the iteration of a rational function they must be either attraction areas parabolic areas or areas of rotation. Areas of rotation are either seal disks or Herman rings with today's designations . The existence of Seal discs was in 1942 by Siegel proved during the existence of Herman rings was open long and only in 1979 by Herman (with the help of a set of Arnold was proven on the solvability of a certain functional equation). ${\displaystyle f}$

literature

• Michael Herman: Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations , Publications Mathématiques de l'IHÉS 49, 5–233 (1979)

Web links

• Arnold-Herman-Ring (Lexicon of Mathematics)