On the residual Julia sets of rational functions

Abstract

We consider the subset of the Julia set called the residual Julia set, which comes from an analogy of the residual limit set of a Kleinian group. We give a necessary and sufficient condition in order that the residual Julia set is empty in the case of hyperbolic rational functions.

Residual Julia sets of rational and transcendental functions

“…Similar results were proved (independently) by Morosawa (see [Mo1] and [Mo2]). One can ask if it is possible to have a rational function with connected Julia set and nonempty residual Julia set.…”

“…Beardon studied this example in his book [Bea1] and he also gave conditions under which the existence of buried components was assured (see [Bea2]). After these results several mathematicians have studied buried components for rational functions (see [BD1] [Mo1], [Mo2] [Q2]). The first discussion about residual Julia sets for functions in class E of finite type was given by Qiao in 1995 [Q1].…”

“…In Section 2 we discussed some results of Morosawa [Mo1] and Qiao [Q2] on conditions for a rational function to have buried points or buried components. In this section we will focus on those results that can be extended to transcendental meromorphic functions.…”

On the dynamics of a family of non-Collet–Eckmann rational maps

“…If all points in some component of the Julia set are buried points, then this component is called a buried component. As we all know, since McMullen [15] gave an example of a rational function with so-called buried points and buried components in its Julia set, there has been much interest in investigating the geometrical and topological properties for buried points and buried components [11,[16][17][18][19][20]. In fact, the existence of buried points and buried components shows that the Julia set has very complex topological properties.…”

Buried points in Julia sets

“…Makienko's conjecture has received attention in the past, with results being limited by topological considerations: see Morosawa [23,24] and Qiao [30] for locally connected Julia sets and non-connected Julia sets, and Sun and Yang [31] for a result similar…”