Cite this article as:
Gumenuk P. A. Siegеl disks and basins of attraction for families of analytic functions. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2005, vol. 5, iss. 1, pp. 12-?.
Siegеl disks and basins of attraction for families of analytic functions
Let be a hyperbolic domain, , let ∆ be a Stolz angle at with respect to the unit disk D, and W a domain containing the point λ0 . Consider an analytic family ; consisting of analytic functions in the domain U with the following expansion , λ ∈ W, for small z. Let be the maximal domain A ⊂ U, such that 0 ∈ A and f l (A) ⊂ A, or the set {0} if there exist no such domains. We prove, that if a sequence converges to λ0 and , then the sequence of the domains converges to 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 we obtain an asymptotic estimate for the size of the neighbourhood V = V (K) of the point λ0 , such that a given compact K ⊂ S lies in A* (0, f l , U) for all .
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