Diagonally self - affine fractal cartoons . Part 3 : “ anomalous ” Hausdorff dimension and multifractal “ localization

H24 Diagonally self-affine fractal cartoons. Part 3: " anomalous " Hausdorff dimension and multifractal " localization " • Illustrated long chapter foreword. An " anomaly " can be dismissed, or welcomed as a challenge to be faced. Moreover, the familiar mantra applies. When facing a challenge, the recommended first step is to become acquainted with its nature as intimately as possible. It is best to create suitable pictures and very literally to " see " and ponder them. This chapter provides a marvelous fresh example. The original was written at Harvard in 1985, with no computer access. Belatedly, this reprint created the need for illustrations and preparing them brought fresh understanding of special examples, turned out to be highly educational, and motivated this lengthy illustrated foreword. In the continuing search for a good definition of self-affine functions, this chapter reveals fresh complexity. It adds a wrinkle to the contrast between generic pure multifractality and the presence of a multifractal mark on the unifractality which is this book's main topic. I have not followed closely the purely mathematical development of the anomalous Hausdorff dimension, but rumor has it that many questions raised by McMullen 1984 remain open despite considerable progress on many fronts and a large literature (including Lalley & Two constructions that are less closely related than they might seem. For reasons described shortly, Figure 1 will be called " nonlocalized, " and Figures 2 and 3 will both be called " localized. " In a first approximation, all three – and also Figures 1 and 2 of Chapter H22 – use the same up, down, 464 FRACTALS IN PHYSICS (TRIESTE, 1985) ♦ ♦ H24 up and up generator. In a second approximation, the arrows along the sticks of this common generator vary from figure to figure. Consider the final functions f(t) that those generators yield by context-free recursion (as defined in Chapter H2, Section 2). First question: Do changes in the arrows' directions suffice to affect f(t)? The answer from pure mathematics is " yes. " To wide surprise, McMullen 1984 found that the localized graph in Figure 2 has the " anomalous " Hausdorff–Besicovitch dimension D HB ∼ 1.45. The non-FIGURE C24-1. In this chapter, the figures illustrate the Foreword. Figures 1, 2, and 3 describe the constructions of three self-affine " cartoons " whose generators are seen on the first three sticks on the upper left panels. …