Dimensions of the Boundary of a Graph-Directed Self-Similar Set With Overlaps
Houston Journal of Mathematics
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.
Deng, Guotai, Chuntai Liu, Sze-Man Ngai.
"Dimensions of the Boundary of a Graph-Directed Self-Similar Set With Overlaps."
Houston Journal of Mathematics, 42 (1): 179-210.