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Studia Mathematica

2002 | 152 | 2 | 105-124

On the Hausdorff dimension of certain self-affine sets

EN

Abstrakty

EN
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.

105-124

wydano
2002

Twórcy

autor
• Mathematical Sciences, University of Liverpool, P.O. Box 147, Liverpool L69 3BX, U.K.
autor
• Mathematical Sciences, University of Liverpool, P.O. Box 147, Liverpool L69 3BX, U.K.