Experimental Mathematics

Dimensions of the Boundaries of Self-Similar Sets

Ka-Sing Lau and Sze-Man Ngai

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We introduce a finite boundary type condition on iterated function systems of contractive similitudes on $\R^d$ Under this condition, we compute the Hausdorff dimension of the boundary of the attractor in terms of the spectral radius of some finite offspring matrix. We describe how to construct such a matrix. We also show that, in this case, the box dimension equals the Hausdorff dimension. In particular, this allows us to compute the Hausdorff dimension of the boundary of a class of self-similar sets defined by expansion matrices with noninteger entries.

Article information

Experiment. Math., Volume 12, Issue 1 (2003), 13-26.

First available in Project Euclid: 29 September 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 28A78: Hausdorff and packing measures
Secondary: 28A80: Fractals [See also 37Fxx]

Self-similar set self-similar tile self-affine tile finite type condition finite boundary type condition Hausdorff dimension box dimension


Lau, Ka-Sing; Ngai, Sze-Man. Dimensions of the Boundaries of Self-Similar Sets. Experiment. Math. 12 (2003), no. 1, 13--26. https://projecteuclid.org/euclid.em/1064858781

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