Pseudotangents to Lipschitz curves

TL;DR

The paper extends prior work on pseudotangents by showing that for any compact, uniformly disconnected $K\subset \mathbb{R}^d$ that admits a Lipschitz capture, there exists a Lipschitz capture $F$ of $K$ with $\Psi-\mathrm{Tan}(F,x)=\mathfrak{C}_U(\mathbb{R}^d;0)$ for all $x\in K$. It also proves a general inclusion $\Psi-\mathrm{Tan}(X,x)\subset \mathfrak{C}_U(\mathbb{R}^d;0)$ for nondegenerate continua, and outlines a constructive strategy to ensure pseudotangent richness at all points of $K$ by successive Lipschitz captures, yielding stronger results than previous work. The work highlights a fundamental difference between tangents and pseudotangents for Lipschitz curves and raises open questions about achieving universal pseudotangents along a single curve. The findings have implications for understanding the geometric structure of Lipschitz images and the complexity of tangent behavior on fractal-like sets.

Abstract

In this paper, we extend the result of arXiv:2409.13662 by showing that the set on which every pseudotangent is obtained on a Lipschitz curve can be any compact, uniformly disconnected set in Euclidean space which admits any Lipschitz capture. We do not obtain a characterization of such sets however, indeed we leave open the very strong question of whether or not a Lipschitz curve can obtain every pseudotangent at every point.

Theorems & Definitions (16)

• Theorem 1.1
• Definition 2.1
• Lemma 2.2: Beer
• Definition 2.3: Tangents, Pseudotangents, BL
• Remark 2.4
• Proposition 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof