The Hausdorff Dimension of Hilbertʼs Coordinate Functions

Abstract

We characterize the coordinate functions of Hilbert's space-filling curve using a directed-graph iterated function system and use this to analyze their fractal properties. In particular, we show that both coordinate functions have graphs of Hausdorff dimension $\frac{3}{2}$ and level sets of dimension $\frac{1}{2}$.