Dimensions of the Boundary of a Graph-Directed Self-Similar Set With Overlaps

Document Type

Article

Publication Date

2016

Publication Title

Houston Journal of Mathematics

Abstract

We set up a framework for computing the Hausdorff and box dimensions of the boundary of each component of a graph self-similar family defined by a graph-directed iterated function system satisfying the so-called graph finite boundary type condition. We show that the boundary of each component has the same Hausdorff and box dimension, with the corresponding Hausdorff measure being positive and σ-finite. These results are natural extensions of existing ones concerning the dimensions of the boundary of a self-similar tile.

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