Hölder curves with exotic tangent spaces

Abstract

An important implication of Rademacher's Differentiation Theorem is that every Lipschitz curve $\Gamma$ infinitesimally looks like a line at almost all of its points in the sense that at ${\mathcal{H}}^1$-almost every point of $\Gamma$, the only tangent to $\Gamma$ is a straight line through the origin. In this article, we show that, in contrast, the infinitesimal structure of Hölder curves can be much more extreme. First we show that for every $s>1$ there exists a $(1/s)$-Hölder curve ${\Gamma }_s$ in a Euclidean space with ${\mathcal{H}}^s({\Gamma }_s)>0$ such that ${\mathcal{H}}^s$-almost every point of ${\Gamma }_s$ admits infinitely many topologically distinct tangents. Second, we study the tangents of self-similar connected sets (which are canonical examples of Hölder curves) and prove that the curves ${\Gamma }_s$ have the additional property that ${\mathcal{H}}^s$-almost every point of ${\Gamma }_s$ admits infinitely many homeomorphically distinct tangents to ${\Gamma }_s$ which are not admitted as (not even bi-Lipschitz to) tangents to any self-similar set at typical points.

43 pages, 4 figures