Outer boundaries of self-similar tiles

Abstract

There are many examples of self-similar tiles that are connected, but whose interior is disconnected. For such tiles we show that the boundary of a component of the interior may be decomposed into a finite union of pieces, each similar to a subset of the outer boundary of the tile. This is significant because the outer boundary typically has lower dimension than the full boundary. We describe a method to realize the outer boundary as the invariant set of a graph-directed iterated function system. The method works under a certain "finiteness'' assumption. While it is not clear that this assumption always holds, and it is problematic to give a rigorous proof that it holds even in cases where it is "visually clear'' that it holds, we give some examples where the method yields clear and nontrivial results. Details concerning the algorithms may be found at the website www.math.cornell.edu/~$sld32/Tiles.html.