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Falconer's formula for the Hausdorff dimension of a self-affine set in R2

Published online by Cambridge University Press:  19 September 2008

Irene Hueter  and
Steven P. Lalley
Irene Hueter
Affiliation:
Purdue University, Department of Statistics, Purdue University, West Lafayette, IN 47907-1399, USA
Steven P. Lalley
Affiliation:
Purdue University, Department of Statistics, Purdue University, West Lafayette, IN 47907-1399, USA
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Extract

Let A1, A2,…,Ak be a finite set of contractive, affine, invertible self-mappings of R2. A compact subset Λ of R2 is said to be self-affine with affinitiesA1, A2,…,Ak if

It is known [8] that for any such set of contractive affine mappings there is a unique (compact) SA set with these affinities. When the affine mappings A1, A2,…,Ak are similarity transformations, the set Λ is said to be self-similar. Self-similar sets are well understood, at least when the images Ai(Λ) have ‘small’ overlap: there is a simple and explicit formula for the Hausdorff and box dimensions [12, 10]; these are always equal; and the δ-dimensional Hausdorff measure of such a set (where δ is the Hausdorff dimension) is always positive and finite.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 1 , February 1995 , pp. 77 - 97
Copyright
Copyright © Cambridge University Press 1995

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References

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