We give a characterization of compact subsets of finite unions of disjoint finite-length curves in ℝⁿ with ω-continuous derivative and without self-intersections. Intuitively, our condition can be formulated as follows: there exists a finite set of regular curves covering a compact set K iff every triple of points of K behaves like a triple of points of a regular curve. This work was inspired by theorems by Jones, Okikiolu, Schul and others that characterize compact subsets of rectifiable or Ahlfors-regular curves. However, their classes of curves are much wider than ours and therefore the condition we obtain and our methods are different.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm211-2-4,
author = {Marcin Pilipczuk},
title = {Characterization of compact subsets of curves with $\omega$-continuous derivatives},
journal = {Fundamenta Mathematicae},
volume = {215},
year = {2011},
pages = {175-195},
zbl = {1215.53008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm211-2-4}
}
Marcin Pilipczuk. Characterization of compact subsets of curves with ω-continuous derivatives. Fundamenta Mathematicae, Tome 215 (2011) pp. 175-195. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm211-2-4/