Connectedness of Julia sets of rational functions
This paper under review is concerned with Julia sets -- the closure of the repelling periodic points. For any rational map of degree two or more, the Julia set is either connected or else has uncountably many connected components. In the polynomial case, all but countably many of these components must be single points. However, this is not true for a rational map. Most Julia sets tend to be complicated fractal subsets of \(\overline{\mathbb{C}}\), except three smooth cases: the circle, the interval, and \(\overline{\mathbb{C}}\). In the theory of (topologically) complex dynamical systems, critical points play an essential role. For the polynomial case, a Julia set is connected iff the orbits of the finite critical points are bounded. But this is not true for a rational map. Indeed, up to the very nice result of Shishikura that any rational map which has one repelling fixed point only has a connected Julia set, almost nothing is known on the connectivity. In Section 2, they give constructive sufficient conditions, as Theorem 2.2, for a basin of attraction to be completely invariant and the Julia set to be connected. Then it is shown that the connectedness of the basin of attraction depends heavily on the fact whether the critical points from the basin tend to the attracting fixed point \(z_0\) via a preimage of \(z_0\) or not. As a consequence they obtain in Section 4 that rational maps with a finite post-critical set or with a Fatou set which contains no Herman rings and each component of which contains at most one critical point, counted without multiplicity, have a connected Julia set. In the last section, they investigate the family \(R(z)= c(z^3-2)/z\): if \(-\frac{\root 3\of 2}{3}\leq c\leq \frac 23\), then the Julia set is connected, otherwise the Cantor set.
scientific article; zbMATH DE number 1833056
scientific article