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

Kevin G. Hare; Nikita Sidorov

Studia Mathematica (2015)

  • Volume: 229, Issue: 3, page 223-232
  • ISSN: 0039-3223

Abstract

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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 | ≥ 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.

How to cite

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Kevin G. Hare, and Nikita Sidorov. "Multidimensional self-affine sets: non-empty interior and the set of uniqueness." Studia Mathematica 229.3 (2015): 223-232. <http://eudml.org/doc/285506>.

@article{KevinG2015,
abstract = {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 $|det M| ≥ 2^\{-1/d\}$, 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.},
author = {Kevin G. Hare, Nikita Sidorov},
journal = {Studia Mathematica},
keywords = {iterated function system; self-affine set; set of uniqueness; Hausdorff dimension},
language = {eng},
number = {3},
pages = {223-232},
title = {Multidimensional self-affine sets: non-empty interior and the set of uniqueness},
url = {http://eudml.org/doc/285506},
volume = {229},
year = {2015},
}

TY - JOUR
AU - Kevin G. Hare
AU - Nikita Sidorov
TI - Multidimensional self-affine sets: non-empty interior and the set of uniqueness
JO - Studia Mathematica
PY - 2015
VL - 229
IS - 3
SP - 223
EP - 232
AB - 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 $|det M| ≥ 2^{-1/d}$, 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.
LA - eng
KW - iterated function system; self-affine set; set of uniqueness; Hausdorff dimension
UR - http://eudml.org/doc/285506
ER -

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