COORDINATE FUNCTIONS OF SPACE FILLING CURVES

Abstract

Given a continuous function X(t) mapping [0, 1] continuously onto [0, 1], several properties are given which this function must fulfill if there is to be another continuous function Y (t) so that (X(t), Y (t)) takes the unit interval onto the unit square. Many of the references to the literature on space-filling curves can be found in [1] where the nondifferentiability and dimension of preimages and graphs of the historical mappings are given. Here, by coordinate functions will be meant continuous functions X(t) and Y (t) which are assumed to take [0, 1] onto [0, 1] so that F (t) = (X(t), Y (t)) takes [0, 1] onto the unit square. Presented here are some properties of such X and Y which will perhaps suggest a characterization. The problem of finding a characterization should involve showing how to construct a function X (t) given Y (t) satisfying certain properties so that the pair form a space-filling curve. It is convenient to consider Y (t) as given and visualize X(t) as ‘sliding’ the x-coordinate in such a way that all the points in the square are covered by (X(t), Y (t)). With this in mind one can first observe the well known fact that for each y ∈ [0, 1], one has Y −1(y) uncountable. Otherwise, it would not be possible for (X(t), Y (t)) to cover the line segment Iy = {(x, y) : x ∈ [0, 1]}. Since Y (t) is continuous, each Y −1(y) is a closed set and consists of at most countably many line segments along with a (possibly empty) nowhere dense perfect set and an at most countable set of points. However, it is not possible to have the line segments cover Y −1(y) by themselves. In fact, a coordinate function for a space-filling curve must have a nowhere dense perfect set in each Y −1(y) which is mapped by F onto Iy = {(x, y) : x ∈ [0.1]}. This will follow from Theorem 1.