Self-similar sets of Hausdorff measure zero and positive packing measure

Abstract

A contractive similarity is a function which preserves the geometry of a object but shrinks it down by some factor. If we have a finite collection of similarities, then there exists a unique compact set K which is the same set as the union of the the images of K under each similarity. This kind of K is called a self-similar set, which is a certain type of fractal. Self-similar sets may satisfy some separation conditions. These conditions describe how much the different similar parts of the self-similar set K may overlap each other. Self-similar sets with some separation condition, such as the open set condition, are understood quite well. However, without any separation conditions the self-similar set may be very complex.

We prove that there exist self-similar sets with Hausdorff measure zero and positive packing measure, in their own dimension. These kinds of sets can be constructed by taking projections of a certain type of self-similar sets on a plane. By taking an orthogonal projection onto a line, we get a new self-similar set which may not satisfy the same separation conditions as the original set on the plane.

It is known that in this kind of projections the Hausdorff dimension is preserved for almost every direction, in the sense of Lebesgue measure. What happens to the corresponding Hausdorff measure is not understood that well. These results help to understand this problem. In addition, the results imply that the packing measure is a better tool for studying some self-similar sets.

This thesis is based on an article ‘Self-similar sets of zero Hausdorff measure and positive packing measure’ written by Peres, Simon and Solomyak, which and published in the year 2000. The most essential parts of this article are presented and explained in this thesis.

Chapter 1 contains preliminary information. In Chapter 2 we introduce one-parameter families of iterated function systems, which lay background for the projections. Main theorems are proved in Chapters 3 and 4. Results concerning the Hausdorff measure are in Chapter 3 and the results concerning the packing measure are in Chapter 4. Finally in Chapter 5 we present two examples.