H\"older curves with exotic tangent spaces

TL;DR

This paper demonstrates that H"older curves can have highly complex tangent structures, with infinitely many topologically distinct tangents at almost every point, contrasting with the linear tangents typical of Lipschitz curves.

Contribution

It constructs explicit examples of H"older curves with extreme tangent behavior and analyzes their properties, revealing new phenomena in geometric measure theory.

Findings

Existence of H"older curves with infinitely many topologically distinct tangents.

Most points on these curves admit infinitely many non-bi-Lipschitz tangent sets.

Such tangent structures are not present in self-similar sets at typical points.

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 H\"older curves can be much more extreme. First we show that for every $s>1$ there exists a $(1/s)$-H\"older 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 H\"older 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.