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
Let be a holomorphic map between Riemann surfaces.
A connected component of the Fatou set is called Herman’s ring if it is a biholomorphic map
on a ring area
with there, so an irrational twist, so
for one is.
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).
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)