\documentclass{rae}
\newtheorem{myth}{Theorem}
\firstpagenumber{1}
\title{THE HAUSDORFF DIMENSION OF HILBERT'S COORDINATE FUNCTIONS}
\author{Mark McClure, Department of Mathematics, UNC at Asheville\\
Asheville, NC 28804, email: {\tt mcmcclur@bulldog.unca.edu}}
\markboth{Santa Barbara Symposium -- M.~McClure}
{Santa Barbara Symposium -- M.~McClure}
\begin{document}\label{zg-conf-mm}
\maketitle
Let $I=[0,1]$ denote the unit interval and let $I^2$ denote the unit
square. Hilbert's space filling curve is a continuous, surjective
function $h:I \rightarrow I^2$ (see \cite{sag}).
The coordinate functions $x$ and $y$
are given by $h(t) = (x(t),y(t))$. Let the graphs of $x$ and $y$ be
denoted by $X$ and $Y$.
This talk focused on the fractal aspects of the sets $X$ and $Y$. In
particular, we showed how to characterize $X$ and $Y$ using a
directed-graph iterated functions system (see \cite{edg} or \cite{mw})
and used this characterization to find the Hausdorff dimensions of
sets associated with $x$ and $y$. The main results are
summarized in the following theorem.
\begin{myth}
The graphs of $x$ and $y$ both have Hausdorff dimension $\frac{3}{2}$
and any non-empty level set of $x$ or $y$ has Hausdorff dimension
$\frac{1}{2}$.
\end{myth}
\begin{thebibliography}{99}
\bibitem[Edg]{edg} G.~A.~Edgar, {\em Measure, Topology, and Fractal
Geometry}. Springer-Verlag, New York, 1990.
\bibitem[MW]{mw} R.~D.~Mauldin and S.~C.~Williams, ``Hausdorff
dimension in graph directed constructions.'' {\em Trans. Amer. Math.
Soc.} {\bf 309} (1988) 811-829.
\bibitem[Sag]{sag} Hans Sagan, {\em Space-Filling Curves}.
Springer-Verlag, New York, 1994.
\end{thebibliography}
\end{document}