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arXiv · 1506.08714

Multidimensional self-affine sets: non-empty interior and the set of uniqueness

Kevin G. Hare·Nikita Sidorov
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Abstract

Let $M$ be a $d\times d$ contracting matrix. In this paper we consider the self-affine iterated function system $\{Mv-u, Mv+u\}$, where $u$ is a cyclic vector. Our main result is as follows: if $|\det M|\ge 2^{-1/d}$, then the attractor $A_M$ has non-empty interior. We also consider the set $\mathcal U_M$ of points in $A_M$ which have a unique address. We show that unless $M$ belongs to a very special (non-generic) class, the Hausdorff dimension of $\mathcal U_M$ is positive. For this special class the full description of $\mathcal U_M$ is given as well. This paper continues our work begun in two previous papers.

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Kevin G. Hare, Nikita Sidorov. 2016-01-11. Multidimensional self-affine sets: non-empty interior and the set of uniqueness. https://doi.org/10.4064/sm8359-1-2016

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