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 $\alpha>0$, there exists a sequence of $\delta$'s converging to zero, $\delta > 0$, and points $x,y \in S^3$ with dist$(x,y) < \delta$, yet dist$(I^h(x), I^h(y)) > \epsilon$, where $\delta^{ -1} = e^{ \left(\frac{ \epsilon^{ -1}}{ \log^{ (1+\alpha)}(\epsilon^{ -1})}\right)}$, and dist is the usual Riemannian distance on $S^3$. In particular, $I^h$ stretches distance much more than a Lipschitz function ($\delta^{ -1} = c\epsilon^{ -1}$) or a H\"{ o}lder function ($\delta^{ -1} = c^\prime(\epsilon^{ -1})^{ p}$, $1 < p < \infty$). Bing's original construction and known alternatives (see text) for $I$ have a modulus of continuity $\delta^{ -1} > c \sqrt{ 2}^{ \epsilon^{ -1}}$, 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 $\delta^{ -1}$ 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 $\delta^{ -1}(\epsilon^{ -1})$ 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.
Categories: math.GT