论文标题
在统一断开的朱莉娅套装上
On uniformly disconnected Julia sets
论文作者
论文摘要
众所周知，双曲合理图的朱莉娅集合在准对称上等同于标准的cantor套件。使用David和Semmes的统一定理，这一结果归结为这样的朱莉娅集合既统一完美又均匀地断开了这一事实。我们研究了$ \ mathbb {s}^n $中的uqr地图的类似问题，$ n \ geq 2 $。引入双曲线UQR地图，我们表明，如果完全断开连接，则此类地图的Julia集将均匀断开。此外，我们表明，如果$ e $是$ \ mathbb {s}^n $设置的紧凑，统一且均匀断开的连接，那么它是柔毛UQR映射$ f：\ m athbb {s}^n \ to \ m athbb {s}^n $ n $ n $ n = 2 = 2 = 2 = 2 = 2 = 2 n = 2的朱莉娅集合。
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 $\mathbb{S}^n$, for $n\geq 2$. 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 if $E$ is a compact, uniformly perfect and uniformly disconnected set in $\mathbb{S}^n$, then it is the Julia set of a hyperbolic UQR map $f:\mathbb{S}^N \to \mathbb{S}^N$ where $N=n$ if $n=2$ and $N=n+1$ otherwise.
