On the dynamics of a family of singularly perturbed rational maps
On the dynamics of a family of singularly perturbed rational maps
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DOI：
10.1016/j.jmaa.2014.10.090
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发表时间：
2015-04
影响因子：
1.3
通讯作者：
Jianxun Fu;Fei Yang
中科院分区：
文献类型：
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作者：
Jianxun Fu;Fei Yang
本文研究了复有理映射族f λ (z)= zn (z2n−λ n+ 1) z2n−λ 3n−1，其中n≥2且λ∈C {λ: λ 2n−2= 1}的动力学行为。如果λ很小，这组有理映射可以看作是单临界多项式z∑zn的扰动。我们证明了f λ的Julia集合J （f λ）是准圆、康托圆集、Sierpiński地毯或退化Sierpiński地毯，只要f λ的自由临界点之一逃逸到原点或无穷远。特别地，我们证明了存在合适的λ使得Julia集合J （f λ）是圆的Cantor集合，但f λ不与它们对应的Julia集合上的任何McMullen映射拓扑共轭。
In this paper, we study the dynamical behavior of the family of complex rational maps which is given by f λ (z)= z n (z 2 n− λ n+ 1) z 2 n− λ 3 n− 1, where n≥ 2 and λ∈ C⁎−{λ: λ 2 n− 2= 1}. This family of rational maps can be seen as a perturbation of the unicritical polynomial z↦ z n if λ is small. We prove that the Julia set J (f λ) of f λ is either a quasicircle, a Cantor set of circles, a Sierpiński carpet or a degenerated Sierpiński carpet provided one of the free critical points of f λ is escaping to the origin or to the infinity. In particular, we prove that there exists suitable λ such that the Julia set J (f λ) is a Cantor set of circles, but f λ is not topologically conjugate to any McMullen map on their corresponding Julia sets.