A survey of complex dimensions, measurability, and the lattice/nonlattice dichotomy
| Autor: | Dettmers, Kristin, Giza, Robert, Knox, Christina, Morales, Rafael, Rock, John A. |
|---|---|
| Rok vydání: | 2015 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | The theory of complex dimensions of fractal strings developed by Lapidus and van Frankenhuijsen has proven to be a powerful tool for the study of Minkowski measurability of fractal subsets of the real line. In a very general setting, the Minkowski measurability of such sets is characterized by the structure of corresponding complex dimensions. Also, this tool is particularly effective in the setting of self-similar fractal subsets of $\mathbb{R}$ which have been shown to be Minkowski measurable if and only if they are nonlattice. This paper features a survey on the pertinent results of Lapidus and van Frankenhuijsen and a preliminary extension of the theory of complex dimensions to subsets of Euclidean space, with an emphasis on self-similar sets that satisfy various separation conditions. This extension is developed in the context of box-counting measurability, an analog of Minkowski measurability, which is shown to be characterized by complex dimensions under certain mild conditions. Comment: 28 pages, 3 figures, submitted to conference proceedings for the Bremen Winter School/Symposium: Diffusion on Fractals and Nonlinear Dynamics. Latest version contains a new abstract, new results, and a shorter section on preliminary material |
| Databáze: | arXiv |
| Externí odkaz: |
|