Abstract:

In this talk we will explain an article of Lukas Geyer "Siegel discs, Herman rings and the Arnold Family" Transaction of the AMS. 353, 3661-2683 (2001). Let $F(z)\; =\; e2\; \pi \; \theta \; iz\; ez$. The main two results of this paper are the following:
(1) F(z) has a Siegel disc at the origin if and only if θ is a Brjuno number.
(2) If θ is number of constant type then the Siegel disc is a quasidisc containing the critical point -1 at the boundary of the Siegel disc.