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

Kevin G. Hare ¹   ; Nikita Sidorov ²

¹ Department of Pure Mathematics University of Waterloo Waterloo, Ontario, Canada N2L 3G1
² School of Mathematics The University of Manchester Oxford Road Manchester M13 9PL, United Kingdom

Studia Mathematica, Tome 229 (2015) no. 3, pp. 223-232

Résumé

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.

Détail

DOI : 10.4064/sm8359-1-2016

Keywords: times real contracting matrix consider self affine iterated function system mv u where cyclic vector main result follows det attractor has non empty interior consider set mathcal points which have unique address unless belongs special non generic class hausdorff dimension mathcal positive special class full description mathcal given paper continues work begun previous papers