Characterization of compact subsets of curves with $\omega$-continuous derivatives

Marcin Pilipczuk¹

¹ Institute of Informatics University of Warsaw Banacha 2 02-097 Warszawa, Poland

Fundamenta Mathematicae, Tome 211 (2011) no. 2, pp. 175-195.

Résumé

We give a characterization of compact subsets of finite unions of disjoint finite-length curves in $\mathbb R^n$ with $\omega$-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.

Détail

DOI : 10.4064/fm211-2-4

Keywords: characterization compact subsets finite unions disjoint finite length curves mathbb omega continuous derivative without self intersections intuitively condition formulated follows there exists finite set regular curves covering compact set every triple points behaves triple points regular curve work inspired theorems jones okikiolu schul others characterize compact subsets rectifiable ahlfors regular curves however their classes curves much wider ours therefore condition obtain methods different