On the Hausdorff dimension of certain self-affine sets

Abercrombie Alex G.., and Nair R.. "On the Hausdorff dimension of certain self-affine sets." Studia Mathematica 152.2 (2002): 105-124. <http://eudml.org/doc/286645>.

@article{AbercrombieAlexG2002,
abstract = {A subset E of ℝⁿ is called self-affine with respect to a collection ϕ₁,...,ϕₜ of affinities if E is the union of the sets ϕ₁(E),...,ϕₜ(E). For S ⊂ ℝⁿ let $Φ(S) = ⋃ _\{1≤j≤t\} ϕ_\{j\}(S)$. If Φ(S) ⊂ S let $E_\{Φ\}(S)$ denote $⋂ _\{k≥0\}\{Φ\}^k(S)$. For given Φ consisting of contracting “pseudo-dilations” (affinities which preserve the directions of the coordinate axes) and subject to further mild technical restrictions we show that there exist self-affine sets $E_\{Φ\}(S)$ of each Hausdorff dimension between zero and a positive number depending on Φ. We also investigate in detail the special class of cases in ℝ², where the images of a fixed square under some maps ϕ₁,...,ϕₜ are some vertical and some horizontal rectangles of sides 1 and 2. This investigation is made possible by an extension of the method of calculating Hausdorff dimension developed by P. Billingsley.},
author = {Abercrombie Alex G.., Nair R.},
journal = {Studia Mathematica},
keywords = {self-affine sets; Hausdorff dimension},
language = {eng},
number = {2},
pages = {105-124},
title = {On the Hausdorff dimension of certain self-affine sets},
url = {http://eudml.org/doc/286645},
volume = {152},
year = {2002},
}

TY - JOUR
AU - Abercrombie Alex G..
AU - Nair R.
TI - On the Hausdorff dimension of certain self-affine sets
JO - Studia Mathematica
PY - 2002
VL - 152
IS - 2
SP - 105
EP - 124
AB - A subset E of ℝⁿ is called self-affine with respect to a collection ϕ₁,...,ϕₜ of affinities if E is the union of the sets ϕ₁(E),...,ϕₜ(E). For S ⊂ ℝⁿ let $Φ(S) = ⋃ _{1≤j≤t} ϕ_{j}(S)$. If Φ(S) ⊂ S let $E_{Φ}(S)$ denote $⋂ _{k≥0}{Φ}^k(S)$. For given Φ consisting of contracting “pseudo-dilations” (affinities which preserve the directions of the coordinate axes) and subject to further mild technical restrictions we show that there exist self-affine sets $E_{Φ}(S)$ of each Hausdorff dimension between zero and a positive number depending on Φ. We also investigate in detail the special class of cases in ℝ², where the images of a fixed square under some maps ϕ₁,...,ϕₜ are some vertical and some horizontal rectangles of sides 1 and 2. This investigation is made possible by an extension of the method of calculating Hausdorff dimension developed by P. Billingsley.
LA - eng
KW - self-affine sets; Hausdorff dimension
UR - http://eudml.org/doc/286645
ER -