#### Date on Master's Thesis/Doctoral Dissertation

8-2011

#### Document Type

Doctoral Dissertation

#### Degree Name

Ph. D.

#### Department

Mathematics

#### Committee Chair

Das, Manabendra Nath

#### Author's Keywords

Fractal geometry; Dimension; Hausdorff measure; Self-similar; Hyperspace; Self-conformal

#### Subject

Fractals; Hausdorff measures; Hyperspace

#### Abstract

Given a metric space (K, d), the hyperspace of K is defined by H(K) = {F c K: F is compact, F ? 0}. H(K) is itself a metric space under the Hausdorff metric dH. Hyperspaces have been extensively studied by topologists since the 1970's, but the measure theoretical study of hyperspaces has lagged, Boardman and Goodey concurrently provided a characterization of a one-parameter family of Hausdorff gauge functions that determine the dimension of H([0, 1]), and this result was extended by McClure to H(X) where X is a self-similar fractal satisfying the Open Set Condition. This dissertation further generalizes these results to include graph-self-similar and self-conformal fractals satisfying the Open Set Condition in Rd. In Chapter 2 it is shown that the dimensions of the underlying fractals may be approximated by the dimensions of sets invariant under particularly constructed subiterated function systems that satisfy the Strong Separation Condition. In Chapter 3, a one-parameter family of gauge functions is constructed which computes the dimensions of the hyperspaces of graph-self-similar sets that satisfy the Strong Separation Condition, after which the approximations of Chapter 2 are applied to extend the result to graph-self-similar sets which satisfy the Open Set Condition. The analogous results for self-conformal sets that satisfy the Open Set Condition are developed in Chapter 4.

#### Recommended Citation

Zapf, Matthew G., "Cantor set approximations and dimension computations in hyperspaces." (2011). *Electronic Theses and Dissertations.* Paper 1632.

http://dx.doi.org/10.18297/etd/1632