# On the Hausdorff dimension of certain self-affine sets

Studia Mathematica (2002)

- Volume: 152, Issue: 2, page 105-124
- ISSN: 0039-3223

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topAbercrombie 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 -

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