ARNOLD DISKS AND THE MODULI OF HERMAN RINGS OF THE 
COMPLEX STANDARD FAMILY

Nuria Fagella (*), CHRISTIAN HENRIKSEN (**)

(*)
Departament de Matematica aplicada i Analisi
Universitat de Barcelona
Gran Via 585
08005 Barcelona
Spain
e-mail: fagella@maia.ub.es

(**)
Department of mathematics
Technical university of Denmark 
Building 303 
DK-2800 Lyngby, Denmark 
e-mail: christian.henriksen@mat.dtu.dk  


ABSTRACT:

We consider the Arnold family of analytic diffeomorphisms of the
circle  $x \mapsto x+ t+ \frac{a}{2\pi} \sin(2\pi x) \,\, \mod(1)$, where
$a,t\in [0,1)$ and its complexification $f_{\l,a}(z)=\l z  \ex^{\frac{a}{2}
(z-\frac{1}{z})}$, with $\l=\ex^{2\pi i t}$a holomorphic self map of $\C^*$. The parameter space contains the well known Arnold tongues $\calt_\a$ for $\a \in [0,1)$ being the rotation number. We are interested in the parameters that belong to the irrational tongues and in particular in those for which the map has  a Herman ring.  
Our goal in this paper is twofold. First we are interested in
studying how the modulus of this Herman ring varies in terms of the
parameter $a$, when $a$ tends to 0 along the curve $\calt_\a$. We survey
the different results that describe this variation including the
complexification of part of the Arnold tongues (called {\em Arnold
disks}) which leads to the best estimate. To work with this complex parameter values we use the concept of the {\em twist coordinate}, a measure of how far from symmetric the Herman rings are. Our second goal is to investigate the slice of parameter space that contains all maps in the family with twist coordinate equal to one half, proving for example that this is a plane in $\C^2$. We show a computer picture of this slice of parameter space and we also present some numerical algorithms that allow us to compute new drawings of non--symmetric Herman rings  of various moduli.