Dimensions of the Boundaries of Self-Similar Sets

Abstract

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.

Keywords:

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