Hölder curves with exotic tangent spaces

An important implication of Rademacher’s Differentiation Theorem is that every Lipschitz curve $\mathrm{\Gamma }$ infinitesimally looks like a line at almost all of its points in the sense that at ${\mathcal{H}}^1$-almost every point of $\mathrm{\Gamma }$, the only tangent to $\mathrm{\Gamma }$ 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 $(1\mathrm{/}s)$-Hölder curve ${\mathrm{\Gamma }}_s$ in a Euclidean space with ${\mathcal{H}}^s({\mathrm{\Gamma }}_s)>0$ such that ${\mathcal{H}}^s$-almost every point of ${\mathrm{\Gamma }}_s$ 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 ${\mathrm{\Gamma }}_s$ have the additional property that ${\mathcal{H}}^s$-almost every point of ${\mathrm{\Gamma }}_s$ admits infinitely many homeomorphically distinct tangents to ${\mathrm{\Gamma }}_s$ which are not admitted as (not even bi-Lipschitz to) tangents to any self-similar set at typical points.