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

Volume 229 / 2015

Kevin G. Hare, Nikita Sidorov Studia Mathematica 229 (2015), 223-232 MSC: Primary 28A80. DOI: 10.4064/sm8359-1-2016 Published online: 15 January 2016


Let $M$ be a $d\times d$ real contracting matrix. 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.


  • Kevin G. HareDepartment of Pure Mathematics
    University of Waterloo
    Waterloo, Ontario, Canada N2L 3G1
  • Nikita SidorovSchool of Mathematics
    The University of Manchester
    Oxford Road
    Manchester M13 9PL, United Kingdom

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