H\'older curves with exotic tangent spaces
| Autor: | Shaw, Eve, Vellis, Vyron |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 weak 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 weak tangents. Second, we study the weak tangents of self-similar connected sets (which are canonical examples of H\"older curves) and prove that infinitely many of the curves $\Gamma_s$ have the additional property that $\mathcal{H}^s$-almost every point of $\Gamma_s$ admits a weak tangent to $\Gamma_s$ which is not admitted as (not even bi-Lipschitz to) a weak tangent to any planar self-similar set at typical points. Comment: 43 pages, 4 figures |
| Databáze: | arXiv |
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