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

Kevin G. Hare; Nikita Sidorov

Studia Mathematica, Tome 231 (2015), p. 223-232 / Harvested from The Polish Digital Mathematics Library

Résumé

Let M be a d × 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 $| d e t M | \geq 2^{- 1 / d}$ , then the attractor $A_{M}$ has non-empty interior. We also consider the set $_{M}$ 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 $_{M}$ is positive. For this special class the full description of $_{M}$ is given as well. This paper continues our work begun in two previous papers.

Publié le : 2015-01-01

Zbl 1334.28020

EUDML-ID : urn:eudml:doc:285506

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     title = {Multidimensional self-affine sets: non-empty interior and the set of uniqueness},
     journal = {Studia Mathematica},
     volume = {231},
     year = {2015},
     pages = {223-232},
     zbl = {1334.28020},
     language = {en},
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Kevin G. Hare; Nikita Sidorov. Multidimensional self-affine sets: non-empty interior and the set of uniqueness. Studia Mathematica, Tome 231 (2015) pp. 223-232. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm8359-1-2016/