On the coordinate functions of Lévy’s dragon curve.

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.