Assouad-type Dimensions of Overlapping Self-affine Sets

TL;DR

This paper investigates the Assouad and quasi-Assouad dimensions of overlapping self-affine sets in the plane, providing new formulas and showing that these dimensions can differ even under strong separation conditions.

Contribution

It introduces a symbolic non-autonomous IFS approach to derive dimension formulas for self-affine sets with overlaps, extending previous results.

Findings

Derived explicit formulas for Assouad and quasi-Assouad dimensions.

Showed that self-affine sets with strong separation can have different Assouad and quasi-Assouad dimensions.

Answered an open question about the difference of these dimensions under strong separation.

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.