Self-affine fractals: Deterministic and random constructions
Abstract
This thesis is a study of dimensional properties of deterministic and random self-affine sets. Strictly self-affine sets (deterministic constructions) are considered in Chapter 1, while Chapter 2 deals with statistically self-affine sets (random constructions). In Chapter 1 we consider self-affine sets generated by affine transformations $S\sb1$, ...,$S\sb{N}$ subject to certain restrictions. The self-affine set is the unique nonempty compact set $\Lambda$ satisfying $\Lambda$ = $\cup\sb{i}S\sb{i}$($\Lambda$). We determine the box and Hausdorff dimensions of such sets and give necessary and sufficient conditions for the box dimension to equal the Hausdorff dimension. These same conditions are also necessary and sufficient for the Hausdorff measure of the set is either zero or infinite. Even though the sets considered in Chapter 1 are deterministic the methods used are largely probabilistic. In particular, we use a continuous projection, mapping sequence space onto the self-affine set under consideration, to define certain measures on the plane, which correspond to distributions of sequences of i.i.d. random vectors (variables) in sequence space. In Chapter 2 we consider a class of statistically self-affine sets in the plane, obtained by a random reccursive Cantor-type construction the first stage of which consists of dividing the unit square into equal rectangles and choosing a random subcollection of them according to some probability distribution F. Under certain conditions on F the limit set is nonempty with positive probability. We then show that, with probability one, the box dimension of the statistically self-affine limit set is equal to a deterministic constant (given that it is nonempty) and determine the value of this constant. A central role in the dimension calculation is played by certain branching processes with random environments.
Degree
Ph.D.
Advisors
Lalley, Purdue University.
Subject Area
Mathematics
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