Hölder curves with exotic tangent spaces
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915497636790272 |
|---|---|
| author | Shaw, Eve Vellis, Vyron |
| author_facet | Shaw, Eve Vellis, Vyron |
| contents | An important implication of Rademacher's Differentiation Theorem is that every Lipschitz curve $Γ$ infinitesimally looks like a line at almost all of its points in the sense that at $\mathcal{H}^1$-almost every point of $Γ$, the only tangent to $Γ$ 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 $Γ_s$ in a Euclidean space with $\mathcal{H}^s(Γ_s)>0$ such that $\mathcal{H}^s$-almost every point of $Γ_s$ 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 $Γ_s$ have the additional property that $\mathcal{H}^s$-almost every point of $Γ_s$ admits infinitely many homeomorphically distinct tangents to $Γ_s$ which are not admitted as (not even bi-Lipschitz to) tangents to any self-similar set at typical points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_13662 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hölder curves with exotic tangent spaces Shaw, Eve Vellis, Vyron Metric Geometry Dynamical Systems 28A80 (Primary) 26A16, 28A75, 53A04 An important implication of Rademacher's Differentiation Theorem is that every Lipschitz curve $Γ$ infinitesimally looks like a line at almost all of its points in the sense that at $\mathcal{H}^1$-almost every point of $Γ$, the only tangent to $Γ$ 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 $Γ_s$ in a Euclidean space with $\mathcal{H}^s(Γ_s)>0$ such that $\mathcal{H}^s$-almost every point of $Γ_s$ 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 $Γ_s$ have the additional property that $\mathcal{H}^s$-almost every point of $Γ_s$ admits infinitely many homeomorphically distinct tangents to $Γ_s$ which are not admitted as (not even bi-Lipschitz to) tangents to any self-similar set at typical points. |
| title | Hölder curves with exotic tangent spaces |
| topic | Metric Geometry Dynamical Systems 28A80 (Primary) 26A16, 28A75, 53A04 |
| url | https://arxiv.org/abs/2409.13662 |