On the composition of polynomials of the form \(z^2+c_n\)
The author defines for a given sequence \((c_n)\) of complex numbers functions \(F_n:= f_n\circ\cdots\circ f_1\), where \(f_k(z):= z^2+ c_k\). Similar to the classical case, a point belongs to the Fatou set if \((F_n)\) is normal in some neighbourhood of \(z\), otherwise to the Julia set. The author deals with the question if these sets have the same properties as the classical sets. If \((c_n)\) is bounded, then, as in the classical case, the Julia set is perfect, nowhere dense and it coincides with the boundary of \(A(\infty)\) which is the component of the Fatou set that contains the point \(\infty\). Furthermore, the author shows that \(F_n\to\infty\) locally uniformly in \(A(\infty)\), but if there is a component \(V\neq A(\infty)\) of the Fatou set, then the set of limits of \((F_n)\) may look much more complicated than in the classical case. If \((c_n)\) is unbounded but \(\log^+| c_n|= O(2^n)\), then the behaviour of \((F_n)\) in the Fatou set is as complicated as in the first case, but the Julia set may even contain interior points, where the Fatou set is never empty since \((F_n)\) is normal in \(\infty\). If \(\log^+| c_n|\neq O(2^n)\), then the point \(\infty\) belongs to the Julia set. The Fatou set may be empty, but if not then \((F_n)\) tends to \(\infty\) locally uniformly in the whole Fatou set. Although the limit-behaviour in the Fatou set becomes easier, the structure of the Julia set becomes more complicated. The author constructs Julia sets which are not perfect (even finite sets occur) or contain interior points without being the whole Riemann sphere.
scientific article; zbMATH DE number 1179890