Sequences of Complex Radius Values that Yield Capture Sierpiński Curve Julia Sets for $n$-Circle Inversion
Keywords:
complex dynamics, Julia set, Sierpiński curve, topologyAbstract
The rational maps $z \mapsto \frac{r^2 z^{n-1}}{z^n-1}$ are related to the geometric action of circle inversion. We prove that for $n$ odd, there exist multiple sequences of radii in parameter space that yield Sierpiński curve Julia sets. Further, although any two such (distinct) radii will yield homeomorphic Julia sets, the dynamics of the functions restricted to their Julia sets are not conjugate.
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