Some dimension results for graphs of continuous functions.

摘要

This dissertation is concerned with the Hausdorff dimension of graphs of continuous functions. In particular, we consider continuous functions whose graphs are 1-dimensional, as well as some alpha-dimensional graphs of functions where alpha ∈ (1, 2). We also introduce the metric dimension and obtain results for both the Hausdorff and metric dimension of graphs of functions.; Given x, y ∈ Rn , let d(x, y) denote the Euclidean distance from x to y. The Hausdorff s-measure of a nonempty bounded subset E of Rn is denoted Hs (E) and is defined by HsE :=lim d→0inf &cubl0;Ui&cubr0; i=1infinity&sqbl0; diam&parl0;Ui&parr0;&sqbr0;s, where diam(Ui) := sup{lcub} d(x, y) : x, y ∈ Ui{rcub} denotes the diameter of Ui, a covering of E is &cubl0;Ui&cubr0;infinityi=1 with i = 1,2,... and diam(Ui) d , and the infimum is taken over all such coverings. The unique number s such that s' s implies Hs' (E) = + infinity and s' > s implies Hs' (E) = 0 is, by definition, the Hausdorff dimension of E.; A method for constructing 1-dimensional functions is shown through two examples. The first construction answers a question posed by Peter Wingren in 1995, and the second modifies a function constructed by Stefan Mazurkiewicz in 1930. Projections onto the x and y-axes of the graph of the modified Mazurkiewicz function are shown to have Hausdorff dimension 1, and it is also shown that a whole class of sets constructed in a similar fashion on the line also have Hausdorff dimension 1.; The Mazurkiewicz function is similar in structure to functions formed using general Sierpinski carpets. Curt McMullen gave a formula for the Hausdorff dimension of these sets in 1984. In order to produce continuous functions from Sierpinski carpets, it is necessary to rotate parts of the generating sets at each iteration. The rotation is either about a line parallel to the x-axis or about a line parallel to the y-axis. It is shown that the two rotations can result in different values for the dimension of the graphs of functions.; The determination of the Hausdorff dimension of a function constructed by Stefan Mazurkiewicz in 1930 was the catalyst for this work, and the question remains open.