Assouad type spectra for some fractal families
Abstract
In a previous paper we introduced a new `dimension spectrum', motivated by the Assouad dimension, designed to give precise information about the scaling structure and homogeneity of a metric space. In this paper we compute the spectrum explicitly for a range of well-studied fractal sets, including: the self-affine carpets of Bedford and McMullen, self-similar and self-conformal sets with overlaps, Mandelbrot percolation, and Moran constructions. We find that the spectrum behaves differently for each of these models and can take on a rich variety of forms. We also consider some applications, including the provision of new bi-Lipschitz invariants and bounds on a family of `tail densities' defined for subsets of the integers.
Keywords
Cite
@article{arxiv.1611.08857,
title = {Assouad type spectra for some fractal families},
author = {Jonathan M. Fraser and Han Yu},
journal= {arXiv preprint arXiv:1611.08857},
year = {2019}
}
Comments
28 pages, 6 figures. This used to be, roughly speaking, the second half of arxiv:1610.02334, which has subsequently been cut in half