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 . If Φ(S) ⊂ S let denote . 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 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm152-2-1,
author = {Abercrombie Alex G.. and Nair R.},
title = {On the Hausdorff dimension of certain self-affine sets},
journal = {Studia Mathematica},
volume = {151},
year = {2002},
pages = {105-124},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm152-2-1}
}
Abercrombie Alex G..; Nair R. On the Hausdorff dimension of certain self-affine sets. Studia Mathematica, Tome 151 (2002) pp. 105-124. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm152-2-1/