Assouad type dimensions and homogeneity of fractals

By Jonathan M. Fraser. 2013

141 citations recorded by the source corpus.

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 quasiself-similar sets and study their relationships with other notions of dimension, such as 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.

Subfields: Condensed Matter Physics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c3092dd5331aacab5b92194e03e373116b37b4cb

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In Condensed Matter Physics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Muckenhoupt Ap-properties of Distance Functions and Applications to Hardy–Sobolev -type Inequalities (2017, result): “We remark that dimA is the “usual” Assouad dimension found in the literature, and we refer to [26] for its basic properties and a historical account and to [10] (and the references therein) for more recent results related to the (upper) Assouad dimension.”
On the Assouad dimension of self-similar sets with overlaps (2014, methodology): “Without relying on the sophisticated notion of Ahlfors regularity, simple direct proofs are also possible, see for example Olson [25], Olsen [24], Henderson [13], and Fraser [10].”
Assouad Dimension and Fractal Geometry (2020, methodology): “Weak tangents were also used in [88] to bound the lower dimension from above.”
New dimension spectra: finer information on scaling and homogeneity (2016, methodology): “iscussion in Section 9 at the end of the paper for more details on this point. The lower dimension, introduced by Larman [L1, L2], is the natural dual of the Assouad dimension. We refer the reader to [Fr] for an in-depth discussion of the relationships and dierences 4 J. M. FRASER AND H. YU between ”