Let $F_\lambda(z)= \lambda z + {\cal O}(z^2)$ be a one parameter holomorphic family of holomorphic maps defined in a neighborhood of $\lambda_0* \overline{\mbox{D}}$. Assume that $F_{\lambda_0}$ has a Siegel disc $\Delta_0$, then the orbit of a point in the interior of the Siegel disc can be followed very closely by periodic orbits of nearby maps. The same technique is applied to Herman rings and Cremer points.