Real Analysis Exchange

On the coordinate functions of Lévy’s dragon curve.

Pieter C. Allaart and Kiko Kawamura

Full-text: Open access

Abstract

Lévy's dragon curve [P. Lévy, Les courbes planes ou gauches et les surfaces composées de parties semblables au tout, J. Ecole Polytechn., 227-247, 249-291 (1938)] is a well-known self-similar planar curve with non-empty interior. We derive an arithmetic expression for the coordinate functions of Lévy's dragon curve, and show that the 3/2 -dimensional Hausdorff measure of the graph of each coordinate function is strictly positive and finite. This complements known dimensional results concerning the coordinate functions of space-filling curves of Peano and Hilbert. The proof is based on deriving suitable uniform upper bounds for the sizes of the graphs' level sets.

Article information

Source
Real Anal. Exchange, Volume 31, Number 1 (2005), 295-308.

Dates
First available in Project Euclid: 5 June 2006

Permanent link to this document
https://projecteuclid.org/euclid.rae/1149516817

Mathematical Reviews number (MathSciNet)
MR2218845

Zentralblatt MATH identifier
1111.28006

Subjects
Primary: 26A27: Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives

Keywords
Levy's dragon curve coordinate function Hausdorff dimension

Citation

Allaart, Pieter C.; Kawamura, Kiko. On the coordinate functions of Lévy’s dragon curve. Real Anal. Exchange 31 (2005), no. 1, 295--308. https://projecteuclid.org/euclid.rae/1149516817


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