Assouad type dimensions and homogeneity of fractals

Abstract

We investigate several aspects of the Assouad dimension and the lower dimension, which together form a natural `dimension pair'. In particular, we compute these dimensions for certain classes of self-affine sets and quasi-self-similar sets and study their relationships with other notions of dimension, like the Hausdorff dimension for example. We also investigate some basic properties of these dimensions including their behaviour regarding unions and products and their set theoretic complexity.

40 pages, 6 figures

References in corpus (2)

• On self-similar sets with overlaps and inverse theorems for entropy
• Dimensions, Whitney covers, and tubular neighborhoods

Cited by in corpus (8)

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• Measures with predetermined regularity and inhomogeneous self-similar sets
• Spherical maximal functions and fractal dimensions of dilation sets
• Assouad spectrum thresholds for some random constructions
• Badly approximable points on self-affine sponges and the lower Assouad dimension
• On the Hausdorff dimension of microsets
• Quantifying inhomogeneity in fractal sets
• Assouad dimensions of Moran sets and Cantor-like sets