Infinitely renormalizable quadratic polynomials, with non-locally connected Julia set
The author presents two methods for producing connected but non-locally connected Julia sets \(J_c\) of infinitely renormalizable quadratic polynomials \(P_c(z)=z^2+c\) for \(c \in \mathbb C\). The main tool for the construction are parabolic perturbations which are defined as follows. Given a root \(c_1\) of a hyperbolic component \(H_1\) of the Mandelbrot set \(M\) such that \(H_1\) is a satellite of another hyperbolic component \(H_0\) of \(M\). Then a first order satellite type parabolic perturbation of \(c_1\) is a perturbation to some special nearby root \(c_2 \in H_1\) of a hyperbolic component \(H_2\). Now, an infinite order satellite type parabolic perturbation \(c_\infty\) is a limit of an infinite sequence of first order satellite type parabolic perturbations. If the boundary of a satellite is parametrized by the internal arguments it carries an orientation. Therefore, it is possible to distinguish perturbations to the left and to the right. An infinite order parabolic perturbation is called one-sided if all first order perturbations are either to the left or to the right, and it is called two-sided if the perturbations are alternating. In the one-sided case the author arranges the perturbations in such a way that the Julia set of the limiting parameter has an external ray which accumulates at two distinct symmetric points. Then one may think of the Julia set as containing a ``topologists spiral''. For two-sided perturbations the author obtains the weaker result that the impression of the interesting external ray contains two symmetric points but accumulation can be guaranteed only at one of them. Then one may have a ``topologists sine'' in mind. Let \(\alpha(c)\) denote the least repelling fixed point of \(P_c\), and let \(\beta(c)\) be the other fixed point. A spine of \(P_c\) is an arc in the filled Julia set of \(P_c\) joining \(\beta(c)\) and \(-\beta(c)\). In a more general setting, containing both of the above cases, the author proves the existence of Julia sets \(J_c\) which do not have a spine. In particular, they are non-arc-wise connected. These seem to be the first examples of connected but not arc-wise connected Julia sets of polynomials. Furthermore, the author shows that his methods cover all known examples of connected but non-locally connected Julia sets of quadratic polynomials that are infinitely renormalizable of satellite type. From this result it follows that the critical point is not accessible. Finally, the author's methods allow to provide examples of Julia sets exhibiting any collection of the mentioned pathologies at the same time.
scientific article; zbMATH DE number 1542068