Assouad-type dimensions of overlapping self-affine sets


  • Jonathan M. Fraser University of St Andrews, Mathematical Institute
  • Alex Rutar University of St Andrews, Mathematical Institute


Assouad dimension, self-affine, quasi-Assouad dimension, symbolic slice


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.



How to Cite

Fraser, J. M., & Rutar, A. (2024). Assouad-type dimensions of overlapping self-affine sets. Annales Fennici Mathematici, 49(1), 3–21.