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Upper and Lower Densities of Cantor Sets using Blanketed Hausdorff Functions

McCoy, Ted
http://rave.ohiolink.edu/etdc/view?acc_num=osu1038902284

Abstract Details

2002, Doctor of Philosophy, Ohio State University, Mathematics.
Suitable gauge functions are used to define three classes of fractal measures: Hausdorff measures, packing measures, and covering measures. Relationships between these fractal measures and corresponding density functions then allow us to bound these measures on fractal sets. These density functions depend not only on the fractal measures, but also on a probability measure. Given a modified Cantor set, the balanced probability measure is the one most naturally determined by this set, but we consider other, unbalanced probability measures as well. In Chapter 2 we study balanced density functions, first by looking at endpoints and then by studying properties of more typical points. We prove a lower density result which allows us to evaluate packing measures of modified Cantor sets, and we prove a weaker upper density result which allows us to evaluate covering measures of such sets subject to an additional condition. We also look at properties of intervals on which upper density is attained, and we use some of these properties to develop a procedure for finding upper density even when this additional condition fails. We then use this procedure to study some of the factors which tend to complicate upper density results. Finally, in Chapter 3, we take a look at unbalanced density functions and study the possibility that useful density results may be possible for some gauge function. We determine the gauge function which provides the best possible results and then show that even this gauge function proves inadequate.
Gerald Edgar (Advisor)
Cantor Sets; Blanketed Hausdorff Functions

Recommended Citations

Citations

  • McCoy, T. (2002). Upper and Lower Densities of Cantor Sets using Blanketed Hausdorff Functions [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1038902284

    APA Style (7th edition)

  • McCoy, Ted. Upper and Lower Densities of Cantor Sets using Blanketed Hausdorff Functions. 2002. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1038902284.

    MLA Style (8th edition)

  • McCoy, Ted. "Upper and Lower Densities of Cantor Sets using Blanketed Hausdorff Functions." Doctoral dissertation, Ohio State University, 2002. http://rave.ohiolink.edu/etdc/view?acc_num=osu1038902284

    Chicago Manual of Style (17th edition)

osu1038902284
986
© 2002, all rights reserved.
This open access ETD is published by The Ohio State University and OhioLINK.