Carathéodory convergence of immediate basins of attraction to a Siegel disk

Let fn be a sequence of analytic functions in a domain U with a common attracting fixed point z0. Suppose that fn converges to f0 uniformly on each compact subset of U and that z0 is a Siegel point of f0. We establish a sufficient condition for the immediate basins of attraction A∗ (z0, fn, U) to form a sequence that converges to the Siegel disk of f0 as to the kernel w. r. t. z0. The same condition is shown to imply the convergence of the Koenigs functions associated with fn to that of f0. Our method allows us also to obtain a kind of quantitative result for analytic one-parametric families.