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# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

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

### Tom 229 / 2015

Studia Mathematica 229 (2015), 223-232 MSC: Primary 28A80. DOI: 10.4064/sm8359-1-2016 Opublikowany online: 15 January 2016

#### Streszczenie

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.

#### Autorzy

• Kevin G. HareDepartment of Pure Mathematics
University of Waterloo
e-mail
• Nikita SidorovSchool of Mathematics
The University of Manchester