Selfsimilar sets of Hausdorff measure zero and positive packing measure 

Author:  Pöyhtäri, Tuomas^{1} 
Organizations: 
^{1}University of Oulu, Faculty of Science, Department of Mathematical Sciences, Mathematics 
Format:  ebook 
Version:  published version 
Access:  open 
Online Access:  PDF Full Text (PDF, 0.6 MB) 
Persistent link:  http://urn.fi/URN:NBN:fi:oulu201303061082 
Language:  English 
Published: 
Oulu :
T. Pöyhtäri,
2013

Publish Date:  20130306 
Physical Description: 
43 p. 
Thesis type:  Master's thesis 
Tutor: 
Järvenpää, Esa 
Reviewer: 
Järvenpää, Esa Suomala, Ville 
Description: 
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 selfsimilar set, which is a certain type of fractal. Selfsimilar sets may satisfy some separation conditions. These conditions describe how much the different similar parts of the selfsimilar set K may overlap each other. Selfsimilar sets with some separation condition, such as the open set condition, are understood quite well. However, without any separation conditions the selfsimilar set may be very complex. We prove that there exist selfsimilar 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 selfsimilar sets on a plane. By taking an orthogonal projection onto a line, we get a new selfsimilar 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 selfsimilar sets. This thesis is based on an article ‘Selfsimilar 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 oneparameter 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. see all

Subjects:  
Copyright information: 
© Tuomas Pöyhtäri, 2013. This publication is copyrighted. You may download, display and print it for your own personal use. Commercial use is prohibited. 