Assouad-type dimensions of overlapping self-affine sets
Abstract
We study the Assouad and quasi-Assoaud dimensions of dominated rectangular self-affine sets in the plane. In contrast to previous work on the dimension theory of self-affine sets, we assume that the sets satisfy certain separation conditions on the projection to the principal axis, but otherwise have arbitrary overlaps in the plane. We introduce and study regularity properties of a certain symbolic non-autonomous iterated function system corresponding to ``symbolic slices'' of the self-affine set. We then establish dimensional formulas for the self-affine sets in terms of the dimension of the projection along with the maximal dimension of slices orthogonal to the projection. Our results are new even in the case when the self-affine set satisfies the strong separation condition: in fact, as an application, we show that self-affine sets satisfying the strong separation condition can have distinct Assouad and quasi-Assouad dimensions, answering a question of the first named author.
Citation
Fraser , J & Rutar , A 2024 , ' Assouad-type dimensions of overlapping self-affine sets ' , Annales Fennici Mathematici , vol. 49 , no. 1 , pp. 3-21 . https://doi.org/10.54330/afm.142535
Publication
Annales Fennici Mathematici
Status
Peer reviewed
ISSN
2737-0690Type
Journal article
Description
Funding: MF was supported by Leverhulme Trust Research Project Grant (RPG-2019-034) and RSE Sabbatical Research Grant (70249). AR was sup-ported by EPSRC Grant EP/V520123/1 and the Natural Sciences and Engineering Research Council of Canada.Collections
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