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arXiv · 1611.08857

Assouad type spectra for some fractal families

Jonathan M. Fraser·Han Yu
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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.

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Jonathan M. Fraser, Han Yu. 2016-11-27. Assouad type spectra for some fractal families. https://arxiv.org/abs/1611.08857

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