论文标题
BING涉及的几何形状
The Geometry of the Bing Involution
论文作者
论文摘要
1952年，Bing出版了3秒$ s^3 $的狂野（不是拓扑结合到平滑的）$ i $。但是在分析上，它到底有多狂野？我们证明，从拓扑结合到$ i $的任何互动$ i^h $都必须具有连续性的几乎指数模量。具体而言，考虑到任何$α> 0 $，存在$δ$的序列，$δ> 0 $，点$ x，y \ in s^3 $，带有dist $（x，y）<δ$，但dist $（i^h（i^h（x），i^h（y），i^h（y），i^h（y）） e^{\ left（\ frac {ε^{ - 1}} {\ log^{（1+α）}（ε^{ - 1}）} \ right）} $，dist是$ s^3 $的常规riemannian距离。尤其是，$ i^h $伸展距离要比Lipschitz函数（$δ^{ - 1} =cε^{ - 1} $）或hölder函数（$δ^{ - 1} = c^\ prime（c^\ prime（ε^{ - 1}） Bing的原始结构和已知替代方案（请参见文本），$ i $具有连续性的模量$δ^{ - 1}> c \ sqrt {2}^{ε^{ - 1}} $，因此定理是相当紧密的 - 我们必须表明，至少必须完全实现polyential expectiential offectepential expectient to polylog，而事实是事实。实际上，可以选择$δ^{ - 1} $的功能，可以选择比此处所述的指数略接近指数（请参见Theorem 1）。使用相同的技术，我们分析了一大批``'''bing的参与，并表明，作为一项学者，它们可以将任何功能$ f：\ mathbb {r}^+ \ rightarrow \ rightarrow \ mathbb {r}^+ $，无论其增长如何快速，我们都可以找到$ j $ j $ j $ j $ j $ j $ j^$ j^$ j^$ j^$ j^$ j^$ j^$ j.连续性的模量$δ^{ - 1}（ε^{ - 1}）$生长速度快于$ f $（近无限）。有关于固有的可不同性（文本中的参考文献）的文献，但据作者知道，连续性固有模量的主题是新的。
In 1952 Bing published a wild (not topologically conjugate to smooth) involution $I$ of the 3-sphere $S^3$. But exactly how wild is it, analytically? We prove that any involution $I^h$, topologically conjugate to $I$, must have a nearly exponential modulus of continuity. Specifically, given any $α>0$, there exists a sequence of $δ$'s converging to zero, $δ> 0$, and points $x,y \in S^3$ with dist$(x,y) < δ$, yet dist$(I^h(x), I^h(y)) > ε$, where $δ^{-1} = e^{\left(\frac{ε^{-1}}{\log^{(1+α)}(ε^{-1})}\right)}$, and dist is the usual Riemannian distance on $S^3$. In particular, $I^h$ stretches distance much more than a Lipschitz function ($δ^{-1} = cε^{-1}$) or a Hölder function ($δ^{-1} = c^\prime(ε^{-1})^{p}$, $1 < p < \infty$). Bing's original construction and known alternatives (see text) for $I$ have a modulus of continuity $δ^{-1} > c \sqrt{2}^{ε^{-1}}$, so the theorem is reasonably tight -- we prove the modulus must be at least exponential up to a polylog, whereas the truth may be fully exponential. Actually, the functional for $δ^{-1}$ coming out of the proof can be chosen slightly closer to exponential than stated here (see Theorem 1). Using the same technique we analyze a large class of ``ramified'' Bing involutions and show, as a scholium, that given any function $f: \mathbb{R}^+ \rightarrow \mathbb{R}^+$, no matter how rapid its growth, we can find a corresponding involution $J$ of the 3-sphere such that any topological conjugate $J^h$ of $J$ must have a modulus of continuity $δ^{-1}(ε^{-1})$ growing faster than $f$ (near infinity). There is a literature on inherent differentiability (references in text) but as far as the authors know the subject of inherent modulus of continuity is new.
