The purpose of this paper is to construct sets, measures and energy forms of certain mixed nested fractals which are spatially homogeneous but not strictly self-similar. We start with the constructions of regular nested fractals, such as Sierpinski gaskets and Koch curves, by employing the iterated map system. Then we show that under the open set condition, the unique invariant (self-similar) measure consists with the normalized Hausdorff measure ristricted on the invariant set. The energy forms construced on regular Sierpinski gaskets and Koch curves is also proved to be a closed form. Next, we use the similar idea, by extending the iterated maps system into a general case, to construct the mixture sets, as well as measures and energy forms. It can be seen that the elements so constructed will not have any strict self-similarity, but them indeed satisfy some weak self-similar properties.
Worcester Polytechnic Institute
All authors have granted to WPI a nonexclusive royalty-free license to distribute copies of the work. Copyright is held by the author or authors, with all rights reserved, unless otherwise noted. If you have any questions, please contact firstname.lastname@example.org.
Liang, Haodong, "On the Constructions of Certain Fractal Mixtures" (2009). Masters Theses (All Theses, All Years). 363.
Sierpinski gasket, energy forms, fractal mixture, self-similar