In 1952, Bing published a wild (not topologically smooth conjugate) degenerate $I$ of the 3-sphere $S^3$. But analytically, exactly how wild is that? To do. Specifically, for a sequence of $\delta$s with constant $c > 1.167$ that converges to zero, $\delta$s > 0$, dist$( x,y) < \delta$ ですが、dist$(I^h(x), I^h(y)) > \epsilon$, where $\delta^{-1} = c^{\sqrt{\epsilon ^{-1}}}$, dist is the usual Riemannian distance on $S^3$. In particular, $I^h$ extends the distance much more than the Lipschitz function ($\delta^{-1} = c^\prime\ \epsilon^{-1}$) or the H\”{o}lder function. ($\delta^{-1} = c^{\prime \prime} (\epsilon^{-1})^{p}$, $1 < p < \infty$). Bing のオリジナルの構造と既知の代替 (参照text) $I$ の場合、連続係数 $\delta^{-1} > c \sqrt{2}^{\epsilon^{-1}}$ so the theorem is rather rigorous — prove the coefficient The truth may be exponential, but it must be at least weakly exponential Using the same technique, we analyze a large class of “branched” Bing degeneracy and, given an arbitrary function $f, ^+ \rightarrow \mathbb{R}^+$, its No matter how fast the growth is, we can find the corresponding involution $J$ in the 3-sphere such that the topological conjugate $J^h$ of $J$ must have the continuity coefficient $\delta^{ -1}(\epsilon^{-1})$ grows faster than $f$ (nearly infinite). There is literature on intrinsic differentiability (reference in text). As the author knows, the subject of inherent continuous coefficients is new.

Dedicated to RH Bing’s life’s work on the 70th anniversary of his discharge from the hospital.