Siegеl disks and basins of attraction for families of analytic functions

P. A. Gumenuk

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 5 (2005) no. 1, pp. 12-25

Résumé

Let $\mathcal{U}\ni 0$ be a hyperbolic domain, $\alpha\in \mathbb{R}\setminus\mathbb{Q}$, let $\Delta$ be a Stolz angle at $\lambda_0=e^{2\pi \alpha}$ with respect to the unit disk $\mathcal{D}$, and $\mathcal{W}$ a domain containing the point $\lambda_0$. Consider an analytic family $f: \mathcal{W}\times \mathcal{U}\to\mathbb{C}$; $(\lambda,z)\mapsto f_\lambda(z)$ consisting of analytic functions in the domain $\mathcal{U}$ with the following expansion $f_\lambda(z)=\lambda z+a_2(\lambda)z^2+\dots$, $\lambda\in \mathcal{W}$, for small $z$. Let $\mathcal{A}^*(0,f_\lambda,\mathcal{U})$ be the maximal domain $\mathcal{A}\subset\mathcal{U}$, such that $0\in \mathcal{A}$ and $f_\lambda(\mathcal{A})\subset \mathcal{A}$, or the set $\{0\}$ if there exist no such domains. We prove, that if a sequence $\{\lambda_n\in \mathcal{W}\cap\Delta\}_{n\in \mathbb{N}}$ converges to $\lambda_0$ and $\mathcal{S}=\mathcal{A}^*(0,f_{\lambda_n},\mathcal{U})\ne\{0\}$, then the sequence of the domains $\mathcal{A}^*(0,f_{\lambda_n},\mathcal{U})$ converges to $\mathcal{S}$ as to the kernel. An example shows, that the analogous statement for convergence with respect to the Hausdorff metric does not hold. In the case $\mathcal{S}\subset \mathcal{U}$ we obtain an asymptotic estimate for the size of the neighbourhood $\mathcal{V}=\mathcal{V}(K)$ of the point $\lambda_0$, such that a given compact $K\subset \mathcal{S}$ lies in $\mathcal{A}^*(0, f_\lambda, \mathcal{U})$ for all $\lambda\in \mathcal{V}\cap\Delta$.