On uniformly disconnected Julia sets
On uniformly disconnected Julia sets
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DOI：
10.1007/s00209-021-02699-6
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发表时间：
2020-04
影响因子：
0.8
通讯作者：
A. Fletcher;Vyron Vellis
中科院分区：
文献类型：
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作者：
A. Fletcher;Vyron Vellis
众所周知，双曲有理映射的Julia集与标准Cantor集是拟对称等价的。利用David和Semmes的一致化定理，这个结果可以归结为这样一个Julia集合既是一致完美的又是一致不连通的。我们研究了UQR映射的Julia集的类似问题。引入双曲UQR映射，证明了这种映射的Julia集合在完全不连通的情况下是一致不连通的。此外，我们还证明了如果i是一个紧致的、一致完美的、一致不连通的集合，那么它就是一个双曲UQR映射的Julia集合，反之亦然。
It is well-known that the Julia set of a hyperbolic rational map is quasisymmetrically equivalent to the standard Cantor set. Using the uniformization theorem of David and Semmes, this result comes down to the fact that such a Julia set is both uniformly perfect and uniformly disconnected. We study the analogous question for Julia sets of UQR maps in, for. Introducing hyperbolic UQR maps, we show that the Julia set of such a map is uniformly disconnected if it is totally disconnected. Moreover, we show that ifEis a compact, uniformly perfect and uniformly disconnected set in, then it is the Julia set of a hyperbolic UQR mapwhereifandotherwise.