SURGERY ON HERMAN RINGS OF THE COMPLEX STANDARD FAMILY Nuria Fagella*, Lukas Geyer *Departament de Matematica aplicada i Analisi Universitat de Barcelona Gran Via 585 08005 Barcelona Spain e-mail: fagella@maia.ub.es ABSTRACT: We consider the standard family (or Arnold family) of circle maps given by $f_{\alpha, \beta}(x)=x + \alpha + \beta \sin(x) \pmod{2\pi}$, for $x,\alpha\in [0,2\pi)$, $\beta \in (0,1)$, and its complexification $F_{\alpha, \beta}(z)=z e^{i\alpha} e^{\frac{\beta}2(z-\frac{1}{z})}$. If $f_{\alpha, \beta}$ is analytically linearizable, there is a Herman ring around the unit circle in the dynamical plane of $F_{\alpha, \beta}$. Given an irrational rotation number $\theta$, the parameters $(\alpha,\beta)$ such that $f_{\alpha, \beta}$ has rotation number $\theta$ form a curve $T_\theta$ in the parameter plane. Using quasi-conformal surgery of the simplest type, we show that if $\theta$ is a Brjuno number, the curve $T_\theta$ can be parametrised real-analytically by the modulus of the Herman ring, from $\beta=0$ up to a point $(\alpha_0,\beta_0)$ with $\beta_0 \leq 1$, for which the Herman ring collapses. Using a result of Herman in [H] and a construction in [BD] we show that the case $\beta_0 <1$ actually exists for some $\theta \in {\mathcal B} \setminus {\mathcal H}$ and, in that case, the boundary of the Herman rings with the corresponding rotation number have two connected components which are quasi-circles, and do not contain any critical point. The same is true for rotation numbers of constant type, although the boundary contains both critical points of $F_{\alpha,\beta}$. [H] M.~Herman, Conjugaison quasi-symm\`etrique des diff\'eomorphismes du cercle \`a des rotations et applications aux disques singuliers de Siegel I. {\em Manuscript}. [BD] I.~N.~Baker and P.~Dom\'{\i}nguez, Analytic self-Maps of the punctured plane, {\em Complex Variables} {\bf 37} (1998), 67--98.