Vertices of Self-Similar Tiles

The set Vn of n-vertices of a tile T in \Rd is the common intersection of T with at least n of its neighbors in a tiling determined by T. Motivated by the recent interest in the topological structure as well as the associated canonical number systems of self-similar tiles, we study the structure of Vn for general and strictly self-similar tiles. We show that if T is a general self-similar tile in \R2 whose interior consists of finitely many components, then any tile in any self-similar tiling generated by T has a finite number of vertices. This work is also motivated by the efforts to understand the structure of the well-known L\\u27evy dragon. In the case T is a strictly self-similar tile or multitile in \Rd, we describe a method to compute the Hausdorff and box dimensions of Vn. By applying this method, we obtain the dimensions of the set of n-vertices of the L\\u27evy dragon for all n≥1