Abstract
Considering a family of rational maps concerning renormalization transformation, we give a perfect description of buried points and phase transition points in the Julia set . Furthermore, we prove that contains an open interval where all points are buried points for some parameters n and λ, which is according to the problem that Curry and Mayer proposed. MSC:37F10, 37F45.
Highlights
It is well known that the Julia set of a rational map is often a fractal
We say that a point in the Julia set is a buried point if it does not lie in the boundary of any Fatou component
If all points in some connected component of the Julia set are buried points, we say this component is called a buried component
Summary
It is well known that the Julia set of a rational map is often a fractal. We say that a point in the Julia set is a buried point if it does not lie in the boundary of any Fatou component. The existence of buried points and buried components shows that the Julia set has very complex topological properties. In this paper, considering the Julia sets J(Tnλ), we prove that the set of buried points may contain an open interval of the real axis R.
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